Tuesday 23 April 2013

Momentum

The sports announcer says, "Going into the all-star break, the Chicago White Sox have the momentum." The headlines declare "Chicago Bulls Gaining Momentum." The coach pumps up his team at half-time, saying "You have the momentum; the critical need is that you use that momentum and bury them in this third quarter."
Momentum is a commonly used term in sports. A team that has the momentum is on the move and is going to take some effort to stop. A team that has a lot of momentum is really on the move and is going to be hard to stop. Momentum is a physics term; it refers to the quantity of motion that an object has. A sports team that is on the move has the momentum. If an object is in motion (on the move) then it has momentum.
Momentum can be defined as "mass in motion." All objects have mass; so if an object is moving, then it has momentum - it has its mass in motion. The amount of momentum that an object has is dependent upon two variables: how much stuff is moving and how fast the stuff is moving. Momentum depends upon the variables mass and velocity. In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object.

Momentum = mass • velocity

In physics, the symbol for the quantity momentum is the lower case "p". Thus, the above equation can be rewritten as

p = m • v

where m is the mass and v is the velocity. The equation illustrates that momentum is directly proportional to an object's mass and directly proportional to the object's velocity.
The units for momentum would be mass units times velocity units. The standard metric unit of momentum is the kg•m/s. While the kg•m/s is the standard metric unit of momentum, there are a variety of other units that are acceptable (though not conventional) units of momentum. Examples include kg•mi/hr, kg•km/hr, and g•cm/s. In each of these examples, a mass unit is multiplied by a velocity unit to provide a momentum unit. This is consistent with the equation for momentum.
Momentum is a vector quantity. As discussed in an earlier unit, a vector quantity is a quantity that is fully described by both magnitude and direction. To fully describe the momentum of a 5-kg bowling ball moving westward at 2 m/s, you must include information about both the magnitude and the direction of the bowling ball. It is not enough to say that the ball has 10 kg•m/s of momentum; the momentum of the ball is not fully described until information about its direction is given. The direction of the momentum vector is the same as the direction of the velocity of the ball. In a previous unit, it was said that the direction of the velocity vector is the same as the direction that an object is moving. If the bowling ball is moving westward, then its momentum can be fully described by saying that it is 10 kg•m/s, westward. As a vector quantity, the momentum of an object is fully described by both magnitude and direction.
From the definition of momentum, it becomes obvious that an object has a large momentum if either its mass or its velocity is large. Both variables are of equal importance in determining the momentum of an object. Consider a Mack truck and a roller skate moving down the street at the same speed. The considerably greater mass of the Mack truck gives it a considerably greater momentum. Yet if the Mack truck were at rest, then the momentum of the least massive roller skate would be the greatest. The momentum of any object that is at rest is 0. Objects at rest do not have momentum - they do not have any "mass in motion." Both variables - mass and velocity - are important in comparing the momentum of two objects.
The momentum equation can help us to think about how a change in one of the two variables might affect the momentum of an object. Consider a 0.5-kg physics cart loaded with one 0.5-kg brick and moving with a speed of 2.0 m/s. The total mass of loaded cart is 1.0 kg and its momentum is 2.0 kg•m/s. If the cart was instead loaded with three 0.5-kg bricks, then the total mass of the loaded cart would be 2.0 kg and its momentum would be 4.0 kg•m/s. A doubling of the mass results in a doubling of the momentum.
Similarly, if the 2.0-kg cart had a velocity of 8.0 m/s (instead of 2.0 m/s), then the cart would have a momentum of 16.0 kg•m/s (instead of 4.0 kg•m/s). A quadrupling in velocity results in a quadrupling of the momentum. These two examples illustrate how the equation p = m•v serves as a "guide to thinking" and not merely a "plug-and-chug recipe for algebraic problem-solving."



Check Your Understanding

Express your understanding of the concept and mathematics of momentum by answering the following questions. Click the button to view the answers.
1. Determine the momentum of a ...
a. 60-kg halfback moving eastward at 9 m/s. b. 1000-kg car moving northward at 20 m/s.
c. 40-kg freshman moving southward at 2 m/s.



2. A car possesses 20 000 units of momentum. What would be the car's new momentum if ...
a. its velocity was doubled. b. its velocity was tripled.
c. its mass was doubled (by adding more passengers and a greater load)
d. both its velocity was doubled and its mass was doubled.
 


Momentum Conservation in Explosions

As discussed in a previous part of Lesson 2, total system momentum is conserved for collisions between objects in an isolated system. For collisions occurring in isolated systems, there are no exceptions to this law. This same principle of momentum conservation can be applied to explosions. In an explosion, an internal impulse acts in order to propel the parts of a system (often a single object) into a variety of directions. After the explosion, the individual parts of the system (that is often a collection of fragments from the original object) have momentum. If the vector sum of all individual parts of the system could be added together to determine the total momentum after the explosion, then it should be the same as the total momentum before the explosion. Just like in collisions, total system momentum is conserved.
Momentum conservation is often demonstrated in a Physics class with a homemade cannon demonstration. A homemade cannon is placed upon a cart and loaded with a tennis ball. The cannon is equipped with a reaction chamber into which a small amount of fuel is inserted. The fuel is ignited, setting off an explosion that propels the tennis ball through the muzzle of the cannon. The impulse of the explosion changes the momentum of the tennis ball as it exits the muzzle at high speed. The cannon experienced the same impulse, changing its momentum from zero to a final value as it recoils backwards. Due to the relatively larger mass of the cannon, its backwards recoil speed is considerably less than the forward speed of the tennis ball.
In the exploding cannon demonstration, total system momentum is conserved. The system consists of two objects - a cannon and a tennis ball. Before the explosion, the total momentum of the system is zero since the cannon and the tennis ball located inside of it are both at rest. After the explosion, the total momentum of the system must still be zero. If the ball acquires 50 units of forward momentum, then the cannon acquires 50 units of backwards momentum. The vector sum of the individual momenta of the two objects is 0. Total system momentum is conserved.
As another demonstration of momentum conservation, consider two low-friction carts at rest on a track. The system consists of the two individual carts initially at rest. The total momentum of the system is zero before the explosion. One of the carts is equipped with a spring-loaded plunger that can be released by tapping on a small pin. The spring is compressed and the carts are placed next to each other. The pin is tapped, the plunger is released, and an explosion-like impulse sets both carts in motion along the track in opposite directions. One cart acquires a rightward momentum while the other cart acquires a leftward momentum. If 20 units of forward momentum are acquired by the rightward-moving cart, then 20 units of backwards momentum is acquired by the leftward-moving cart. The vector sum of the momentum of the individual carts is 0 units. Total system momentum is conserved.


Equal and Opposite Momentum Changes
Just like in collisions, the two objects involved encounter the same force for the same amount of time directed in opposite directions. This results in impulses that are equal in magnitude and opposite in direction. And since an impulse causes and is equal to a change in momentum, both carts encounter momentum changes that are equal in magnitude and opposite in direction. If the exploding system includes two objects or two parts, this principle can be stated in the form of an equation as:
If the masses of the two objects are equal, then their post-explosion velocity will be equal in magnitude (assuming the system is initially at rest). If the masses of the two objects are unequal, then they will be set in motion by the explosion with different speeds. Yet even if the masses of the two objects are different, the momentum change of the two objects (mass • velocity change) will be equal in magnitude.
The diagram below depicts a variety of situations involving explosion-like impulses acting between two carts on a low-friction track. The mass of the carts is different in each situation. In each situation, total system momentum is conserved as the momentum change of one cart is equal and opposite the momentum change of the other cart.

In each of the above situations, the impulse on the carts is the same - a value of 20 kg•cm/s (or cN•s). Since the same spring is used, the same impulse is delivered. Thus, each cart encounters the same momentum change in every situation - a value of 20 kg•cm/s. For the same momentum change, an object with twice the mass will encounter one-half the velocity change. And an object with four times the mass will encounter one-fourth the velocity change.


Solving Explosion Momentum Problems
Since total system momentum is conserved in an explosion occurring in an isolated system, momentum principles can be used to make predictions about the resulting velocity of an object. Problem solving for explosion situations is a common part of most high school physics experiences. Consider for instance the following problem:
A 56.2-gram tennis ball is loaded into a 1.27-kg homemade cannon. The cannon is at rest when it is ignited. Immediately after the impulse of the explosion, a photogate timer measures the cannon to recoil backwards a distance of 6.1 cm in 0.0218 seconds. Determine the post-explosion speed of the cannon and of the tennis ball.
Like any problem in physics, this one is best approached by listing the known information.
Given:
Cannon:m = 1.27 kgd = 6.1 cmt = 0.0218 s
Ball:m = 56.2 g = 0.0562 kg
The strategy for solving for the speed of the cannon is to recognize that the cannon travels 6.1 cm at a constant speed in the 0.0218 seconds. The speed can be assumed constant since the problem states that it was measured after the impulse of the explosion when the acceleration had ceased. Since the cannon was moving at constant speed during this time, the distance/time ratio will provide a post-explosion speed value.
vcannon = d / t = (6.1 cm) / (0.0218 s) = 280 cm/s (rounded)
The strategy for solving for the post-explosion speed of the tennis ball involves using momentum conservation principles. If momentum is to be conserved, then the after-explosion momentum of the system must be zero (since the pre-explosion momentum was zero). For this to be true, then the post-explosion momentum of the tennis ball must be equal in magnitude (and opposite in direction) of that of the cannon. That is,
mball • vball = - mcannon • vcannon
The negative sign in the above equation serves the purpose of making the momenta of the two objects opposite in direction. Now values of mass and velocity can be substituted into the above equation to determine the post-explosion velocity of the tennis ball. (Note that a negative velocity has been inserted for the cannon's velocity.)
(0.0562 kg) • vball = - (1.27 kg) • (-280 cm/s) vball = - (1.27 kg) • (-280 cm/s) / (0.0562 kg)
vball = 6323.26 cm/s
vball = 63.2 m/s
Using momentum explosion, the ball is propelled forward with a speed of 63.2 m/s - that's 141 miles/hour!

It's worth noting that another method of solving for the ball's velocity would be to use a momentum table similar to the one used previously in Lesson 2 for collision problems. In the table, the pre- and post-explosion momentum of the cannon and the tennis ball. This is illustrated below.

MomentumBefore Explosion
MomentumAfter Explosion
Cannon
0
(1.27 kg) • (-280 cm/s) = -355 kg•cm/s
Tennis Ball
0
(0.0562 kg) • v
Total
0
0
The variable v is used for the post-explosion velocity of the tennis ball. Using the table, one would state that the sum of the cannon and the tennis ball's momentum after the explosion must sum to the total system momentum of 0 as listed in the last row of the table. Thus,
-355 kg•cm/s + (0.0562 kg) • v = 0
Solving for v yields 6323 cm/s or 63.2 m/s - consistent with the previous solution method.
Using the table means that you can use the same problem solving strategy for both collisions and explosions. After all, it is the same momentum conservation principle that governs both situations. Whether it is a collision or an explosion, if it occurs in an isolated system, then each object involved encounters the same impulse to cause the same momentum change. The impulse and momentum change on each object are equal in magnitude and opposite in direction. Thus, the total system momentum is conserved.


Newton's First Law

In a previous chapter of study, the variety of ways by which motion can be described (words, graphs, diagrams, numbers, etc.) was discussed. In this unit (Newton's Laws of Motion), the ways in which motion can be explained will be discussed. Isaac Newton (a 17th century scientist) put forth a variety of laws that explain why objects move (or don't move) as they do. These three laws have become known as Newton's three laws of motion. The focus of Lesson 1 is Newton's first law of motion - sometimes referred to as the law of inertia.
Newton's first law of motion is often stated as
An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

There are two parts to this statement - one that predicts the behavior of stationary objects and the other that predicts the behavior of moving objects. The two parts are summarized in the following diagram.

The behavior of all objects can be described by saying that objects tend to "keep on doing what they're doing" (unless acted upon by an unbalanced force). If at rest, they will continue in this same state of rest. If in motion with an eastward velocity of 5 m/s, they will continue in this same state of motion (5 m/s, East). If in motion with a leftward velocity of 2 m/s, they will continue in this same state of motion (2 m/s, left). The state of motion of an object is maintained as long as the object is not acted upon by an unbalanced force. All objects resist changes in their state of motion - they tend to "keep on doing what they're doing."

Suppose that you filled a baking dish to the rim with water and walked around an oval track making an attempt to complete a lap in the least amount of time. The water would have a tendency to spill from the container during specific locations on the track. In general the water spilled when:
  • the container was at rest and you attempted to move it
  • the container was in motion and you attempted to stop it
  • the container was moving in one direction and you attempted to change its direction.
The water spills whenever the state of motion of the container is changed. The water resisted this change in its own state of motion. The water tended to "keep on doing what it was doing." The container was moved from rest to a high speed at the starting line; the water remained at rest and spilled onto the table. The container was stopped near the finish line; the water kept moving and spilled over container's leading edge. The container was forced to move in a different direction to make it around a curve; the water kept moving in the same direction and spilled over its edge. The behavior of the water during the lap around the track can be explained by Newton's first law of motion.

Everyday Applications of Newton's First Law

There are many applications of Newton's first law of motion. Consider some of your experiences in an automobile. Have you ever observed the behavior of coffee in a coffee cup filled to the rim while starting a car from rest or while bringing a car to rest from a state of motion? Coffee "keeps on doing what it is doing." When you accelerate a car from rest, the road provides an unbalanced force on the spinning wheels to push the car forward; yet the coffee (that was at rest) wants to stay at rest. While the car accelerates forward, the coffee remains in the same position; subsequently, the car accelerates out from under the coffee and the coffee spills in your lap. On the other hand, when braking from a state of motion the coffee continues forward with the same speed and in the same direction, ultimately hitting the windshield or the dash. Coffee in motion stays in motion.
Have you ever experienced inertia (resisting changes in your state of motion) in an automobile while it is braking to a stop? The force of the road on the locked wheels provides the unbalanced force to change the car's state of motion, yet there is no unbalanced force to change your own state of motion. Thus, you continue in motion, sliding along the seat in forward motion. A person in motion stays in motion with the same speed and in the same direction ... unless acted upon by the unbalanced force of a seat belt. Yes! Seat belts are used to provide safety for passengers whose motion is governed by Newton's laws. The seat belt provides the unbalanced force that brings you from a state of motion to a state of rest. Perhaps you could speculate what would occur when no seat belt is used.



There are many more applications of Newton's first law of motion. Several applications are listed below. Perhaps you could think about the law of inertia and provide explanations for each application.
  • Blood rushes from your head to your feet while quickly stopping when riding on a descending elevator.
  • The head of a hammer can be tightened onto the wooden handle by banging the bottom of the handle against a hard surface.
  • A brick is painlessly broken over the hand of a physics teacher by slamming it with a hammer. (CAUTION: do not attempt this at home!)
  • To dislodge ketchup from the bottom of a ketchup bottle, it is often turned upside down and thrusted downward at high speeds and then abruptly halted.
  • Headrests are placed in cars to prevent whiplash injuries during rear-end collisions.
  • While riding a skateboard (or wagon or bicycle), you fly forward off the board when hitting a curb or rock or other object that abruptly halts the motion of the skateboard.

Inertia and Mass

Newton's first law of motion states that "An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force." Objects tend to "keep on doing what they're doing." In fact, it is the natural tendency of objects to resist changes in their state of motion. This tendency to resist changes in their state of motion is described as inertia.
Inertia: the resistance an object has to a change in its state of motion.
Newton's conception of inertia stood in direct opposition to more popular conceptions about motion. The dominant thought prior to Newton's day was that it was the natural tendency of objects to come to a rest position. Moving objects, so it was believed, would eventually stop moving; a force was necessary to keep an object moving. But if left to itself, a moving object would eventually come to rest and an object at rest would stay at rest; thus, the idea that dominated people's thinking for nearly 2000 years prior to Newton was that it was the natural tendency of all objects to assume a rest position.

Galileo and the Concept of Inertia

Galileo, a premier scientist in the seventeenth century, developed the concept of inertia. Galileo reasoned that moving objects eventually stop because of a force called friction. In experiments using a pair of inclined planes facing each other, Galileo observed that a ball would roll down one plane and up the opposite plane to approximately the same height. If smoother planes were used, the ball would roll up the opposite plane even closer to the original height. Galileo reasoned that any difference between initial and final heights was due to the presence of friction. Galileo postulated that if friction could be entirely eliminated, then the ball would reach exactly the same height.
Galileo further observed that regardless of the angle at which the planes were oriented, the final height was almost always equal to the initial height. If the slope of the opposite incline were reduced, then the ball would roll a further distance in order to reach that original height.
Galileo's reasoning continued - if the opposite incline were elevated at nearly a 0-degree angle, then the ball would roll almost forever in an effort to reach the original height. And if the opposing incline was not even inclined at all (that is, if it were oriented along the horizontal), then ... an object in motion would continue in motion... .

Forces Don't Keep Objects Moving

Isaac Newton built on Galileo's thoughts about motion. Newton's first law of motion declares that a force is not needed to keep an object in motion. Slide a book across a table and watch it slide to a rest position. The book in motion on the table top does not come to a rest position because of the absence of a force; rather it is the presence of a force - that force being the force of friction - that brings the book to a rest position. In the absence of a force of friction, the book would continue in motion with the same speed and direction - forever! (Or at least to the end of the table top.) A force is not required to keep a moving book in motion. In actuality, it is a force that brings the book to rest.


Mass as a Measure of the Amount of Inertia

All objects resist changes in their state of motion. All objects have this tendency - they have inertia. But do some objects have more of a tendency to resist changes than others? Absolutely yes! The tendency of an object to resist changes in its state of motion varies with mass. Mass is that quantity that is solely dependent upon the inertia of an object. The more inertia that an object has, the more mass that it has. A more massive object has a greater tendency to resist changes in its state of motion.
Suppose that there are two seemingly identical bricks at rest on the physics lecture table. Yet one brick consists of mortar and the other brick consists of Styrofoam. Without lifting the bricks, how could you tell which brick was the Styrofoam brick? You could give the bricks an identical push in an effort to change their state of motion. The brick that offers the least resistance is the brick with the least inertia - and therefore the brick with the least mass (i.e., the Styrofoam brick).
A common physics demonstration relies on this principle that the more massive the object, the more that object resist changes in its state of motion. The demonstration goes as follows: several massive books are placed upon a teacher's head. A wooden board is placed on top of the books and a hammer is used to drive a nail into the board. Due to the large mass of the books, the force of the hammer is sufficiently resisted (inertia). This is demonstrated by the fact that the teacher does not feel the hammer blow. (Of course, this story may explain many of the observations that you previously have made concerning your "weird physics teacher.") A common variation of this demonstration involves breaking a brick over the teacher's hand using the swift blow of a hammer. The massive bricks resist the force and the hand is not hurt. (CAUTION: do not try these demonstrations at home.)


State of Motion

Inertia is the tendency of an object to resist changes in its state of motion. But what is meant by the phrase state of motion? The state of motion of an object is defined by its velocity - the speed with a direction. Thus, inertia could be redefined as follows:
Inertia: tendency of an object to resist changes in its velocity.
An object at rest has zero velocity - and (in the absence of an unbalanced force) will remain with a zero velocity. Such an object will not change its state of motion (i.e., velocity) unless acted upon by an unbalanced force. An object in motion with a velocity of 2 m/s, East will (in the absence of an unbalanced force) remain in motion with a velocity of 2 m/s, East. Such an object will not change its state of motion (i.e., velocity) unless acted upon by an unbalanced force. Objects resist changes in their velocity.
As learned in an earlier unit, an object that is not changing its velocity is said to have an acceleration of 0 m/s/s. Thus, we could provide an alternative means of defining inertia:
Inertia: tendency of an object to resist accelerations.

 

Balanced and Unbalanced Forces

Newton's first law of motion has been frequently stated throughout this lesson.
An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

But what exactly is meant by the phrase unbalanced force? What is an unbalanced force? In pursuit of an answer, we will first consider a physics book at rest on a tabletop. There are two forces acting upon the book. One force - the Earth's gravitational pull - exerts a downward force. The other force - the push of the table on the book (sometimes referred to as a normal force) - pushes upward on the book.
Since these two forces are of equal magnitude and in opposite directions, they balance each other. The book is said to be at equilibrium. There is no unbalanced force acting upon the book and thus the book maintains its state of motion. When all the forces acting upon an object balance each other, the object will be at equilibrium; it will not accelerate. (Note: diagrams such as the one above are known as free-body diagrams and will be discussed in detail in Lesson 2.)
Consider another example involving balanced forces - a person standing upon the ground. There are two forces acting upon the person. The force of gravity exerts a downward force. The floor of the floor exerts an upward force.
Since these two forces are of equal magnitude and in opposite directions, they balance each other. The person is at equilibrium. There is no unbalanced force acting upon the person and thus the person maintains its state of motion. (Note: diagrams such as the one above are known as free-body diagrams and will be discussed in detail in Lesson 2.)
Now consider a book sliding from left to right across a tabletop. Sometime in the prior history of the book, it may have been given a shove and set in motion from a rest position. Or perhaps it acquired its motion by sliding down an incline from an elevated position. Whatever the case, our focus is not upon the history of the book but rather upon the current situation of a book sliding to the right across a tabletop. The book is in motion and at the moment there is no one pushing it to the right. (Remember: a force is not needed to keep a moving object moving to the right.) The forces acting upon the book are shown below.
The force of gravity pulling downward and the force of the table pushing upwards on the book are of equal magnitude and opposite directions. These two forces balance each other. Yet there is no force present to balance the force of friction. As the book moves to the right, friction acts to the left to slow the book down. There is an unbalanced force; and as such, the book changes its state of motion. The book is not at equilibrium and subsequently accelerates. Unbalanced forces cause accelerations. In this case, the unbalanced force is directed opposite the book's motion and will cause it to slow down. (Note: diagrams such as the one above are known as free-body diagrams and will be discussed in detail in Lesson 2.)

To determine if the forces acting upon an object are balanced or unbalanced, an analysis must first be conducted to determine what forces are acting upon the object and in what direction. If two individual forces are of equal magnitude and opposite direction, then the forces are said to be balanced. An object is said to be acted upon by an unbalanced force only when there is an individual force that is not being balanced by a force of equal magnitude and in the opposite direction. Such analyses are discussed in Lesson 2 of this unit and applied in Lesson 3.

Check Your Understanding

Luke Autbeloe drops an approximately 5.0 kg fat cat (weight = 50.0 N) off the roof of his house into the swimming pool below. Upon encountering the pool, the cat encounters a 50.0 N upward resistance force (assumed to be constant). Use this description to answer the following questions. Click the button to view the correct answers.
1. Which one of the velocity-time graphs best describes the motion of the cat? Support your answer with sound reasoning.
 
 
 

Newton's Second Law

Newton's first law of motion predicts the behavior of objects for which all existing forces are balanced. The first law - sometimes referred to as the law of inertia - states that if the forces acting upon an object are balanced, then the acceleration of that object will be 0 m/s/s. Objects at equilibrium (the condition in which all forces balance) will not accelerate. According to Newton, an object will only accelerate if there is a net or unbalanced force acting upon it. The presence of an unbalanced force will accelerate an object - changing its speed, its direction, or both its speed and direction.
Newton's second law of motion pertains to the behavior of objects for which all existing forces are not balanced. The second law states that the acceleration of an object is dependent upon two variables - the net force acting upon the object and the mass of the object. The acceleration of an object depends directly upon the net force acting upon the object, and inversely upon the mass of the object. As the force acting upon an object is increased, the acceleration of the object is increased. As the mass of an object is increased, the acceleration of the object is decreased.

Newton's second law of motion can be formally stated as follows:
The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
This verbal statement can be expressed in equation form as follows:

a = Fnet / m

The above equation is often rearranged to a more familiar form as shown below. The net force is equated to the product of the mass times the acceleration.

Fnet = m * a

In this entire discussion, the emphasis has been on the net force. The acceleration is directly proportional to the net force; the net force equals mass times acceleration; the acceleration in the same direction as the net force; an acceleration is produced by a net force. The NET FORCE. It is important to remember this distinction. Do not use the value of merely "any 'ole force" in the above equation. It is the net force that is related to acceleration. As discussed in an earlier lesson, the net force is the vector sum of all the forces. If all the individual forces acting upon an object are known, then the net force can be determined. If necessary, review this principle by returning to the practice questions in Lesson 2.

Consistent with the above equation, a unit of force is equal to a unit of mass times a unit of acceleration. By substituting standard metric units for force, mass, and acceleration into the above equation, the following unit equivalency can be written.
The definition of the standard metric unit of force is stated by the above equation. One Newton is defined as the amount of force required to give a 1-kg mass an acceleration of 1 m/s/s.

Finding Acceleration

As learned earlier in Lesson 3 (as well as in Lesson 2), the net force is the vector sum of all the individual forces. In Lesson 2, we learned how to determine the net force if the magnitudes of all the individual forces are known. In this lesson, we will learn how to determine the acceleration of an object if the magnitudes of all the individual forces are known. The three major equations that will be useful are the equation for net force (Fnet = m•a), the equation for gravitational force (Fgrav = m•g), and the equation for frictional force (Ffrict = μ•Fnorm).
The process of determining the acceleration of an object demands that the mass and the net force are known. If mass (m) and net force (Fnet) are known, then the acceleration is determined by use of the equation.
Thus, the task involves using the above equations, the given information, and your understanding of Newton's laws to determine the acceleration. To gain a feel for how this method is applied, try the following practice problems. Once you have solved the problems, click the button to check your answers.


Avoid forcing a problem into the form of a previously solved problem. Problems in physics will seldom look the same. Instead of solving problems by rote or by mimicry of a previously solved problem, utilize your conceptual understanding of Newton's laws to work towards solutions to problems. Use your understanding of weight and mass to find the m or the Fgrav in a problem. Use your conceptual understanding of net force (vector sum of all the forces) to find the value of Fnet or the value of an individual force. Do not divorce the solving of physics problems from your understanding of physics concepts. If you are unable to solve physics problems like those above, it is does not necessarily mean that you are having math difficulties. It is likely that you are having a physics concepts difficulty.
 
 
 

Free Fall and Air Resistance

In a previous unit, it was stated that all objects (regardless of their mass) free fall with the same acceleration - 9.8 m/s/s. This particular acceleration value is so important in physics that it has its own peculiar name - the acceleration of gravity - and its own peculiar symbol - g. But why do all objects free fall at the same rate of acceleration regardless of their mass? Is it because they all weigh the same? ... because they all have the same gravity? ... because the air resistance is the same for each? Why? These questions will be explored in this section of Lesson 3.
In addition to an exploration of free fall, the motion of objects that encounter air resistance will also be analyzed. In particular, two questions will be explored:
  • Why do objects that encounter air resistance ultimately reach a terminal velocity?
  • In situations in which there is air resistance, why do more massive objects fall faster than less massive objects?
To answer the above questions, Newton's second law of motion (Fnet = m•a) will be applied to analyze the motion of objects that are falling under the sole influence of gravity (free fall) and under the dual influence of gravity and air resistance.

Free Fall Motion

As learned in an earlier unit, free fall is a special type of motion in which the only force acting upon an object is gravity. Objects that are said to be undergoing free fall, are not encountering a significant force of air resistance; they are falling under the sole influence of gravity. Under such conditions, all objects will fall with the same rate of acceleration, regardless of their mass. But why? Consider the free-falling motion of a 1000-kg baby elephant and a 1-kg overgrown mouse.
If Newton's second law were applied to their falling motion, and if a free-body diagram were constructed, then it would be seen that the 1000-kg baby elephant would experiences a greater force of gravity. This greater force of gravity would have a direct affect upon the elephant's acceleration; thus, based on force alone, it might be thought that the 1000-kg baby elephant would accelerate faster. But acceleration depends upon two factors: force and mass. The 1000-kg baby elephant obviously has more mass (or inertia). This increased mass has an inverse affect upon the elephant's acceleration. And thus, the direct affect of greater force on the 1000-kg elephant is offset by the inverse affect of the greater mass of the 1000-kg elephant; and so each object accelerates at the same rate - approximately 10 m/s/s. The ratio of force to mass (Fnet/m) is the same for the elephant and the mouse under situations involving free fall.
This ratio (Fnet/m) is sometimes called the gravitational field strength and is expressed as 9.8 N/kg (for a location upon Earth's surface). The gravitational field strength is a property of the location within Earth's gravitational field and not a property of the baby elephant nor the mouse. All objects placed upon Earth's surface will experience this amount of force (9.8 N) upon every 1 kilogram of mass within the object. Being a property of the location within Earth's gravitational field and not a property of the free falling object itself, all objects on Earth's surface will experience this amount of force per mass. As such, all objects free fall at the same rate regardless of their mass. Because the 9.8 N/kg gravitational field at Earth's surface causes a 9.8 m/s/s acceleration of any object placed there, we often call this ratio the acceleration of gravity. (Gravitational forces will be discussed in greater detail in a later unit of The Physics Classroom tutorial.)

Newton's Third Law

A force is a push or a pull upon an object that results from its interaction with another object. Forces result from interactions! As discussed in Lesson 2, some forces result from contact interactions (normal, frictional, tensional, and applied forces are examples of contact forces) and other forces are the result of action-at-a-distance interactions (gravitational, electrical, and magnetic forces). According to Newton, whenever objects A and B interact with each other, they exert forces upon each other. When you sit in your chair, your body exerts a downward force on the chair and the chair exerts an upward force on your body. There are two forces resulting from this interaction - a force on the chair and a force on your body. These two forces are called action and reaction forces and are the subject of Newton's third law of motion. Formally stated, Newton's third law is:
For every action, there is an equal and opposite reaction.
The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the forces on the first object equals the size of the force on the second object. The direction of the force on the first object is opposite to the direction of the force on the second object. Forces always come in pairs - equal and opposite action-reaction force pairs.
A variety of action-reaction force pairs are evident in nature. Consider the propulsion of a fish through the water. A fish uses its fins to push water backwards. But a push on the water will only serve to accelerate the water. Since forces result from mutual interactions, the water must also be pushing the fish forwards, propelling the fish through the water. The size of the force on the water equals the size of the force on the fish; the direction of the force on the water (backwards) is opposite the direction of the force on the fish (forwards). For every action, there is an equal (in size) and opposite (in direction) reaction force. Action-reaction force pairs make it possible for fish to swim.
Consider the flying motion of birds. A bird flies by use of its wings. The wings of a bird push air downwards. Since forces result from mutual interactions, the air must also be pushing the bird upwards. The size of the force on the air equals the size of the force on the bird; the direction of the force on the air (downwards) is opposite the direction of the force on the bird (upwards). For every action, there is an equal (in size) and opposite (in direction) reaction. Action-reaction force pairs make it possible for birds to fly.
Consider the motion of a car on the way to school. A car is equipped with wheels that spin. As the wheels spin, they grip the road and push the road backwards. Since forces result from mutual interactions, the road must also be pushing the wheels forward. The size of the force on the road equals the size of the force on the wheels (or car); the direction of the force on the road (backwards) is opposite the direction of the force on the wheels (forwards). For every action, there is an equal (in size) and opposite (in direction) reaction. Action-reaction force pairs make it possible for cars to move along a roadway surface.
 
 
 

Friday 12 April 2013

2.5 FORCE
  Example:

Balanced Force
  When the forces acting on an object are balanced
, they cancel each other out. The net force
 is zero
.  Effect
:  the object at is at rest
 [ velocity = 0]  or moves at constant
velocity
 [ a = 0]

Weight, W = Lift, U    Thrust, F = drag, G 
When the forces acting on an object are not
balanced
, there must be a net force
 acting on it. The net force is known as the unbalanced
force
 or the resultant force.
 
Unbalanced Force/ Resultant Force

Effect : Can cause a body to
-

change it state at rest
 (an object will accelerate
-

change it state of motion
 (a moving object will decelerate or change its direction)
 Force, Mass & Acceleration
 The acceleration produced by  a force on an object is directly proportional
 to the magnitude of the net
force
 applied and is inversely proportional to the mass of the object. The direction of the acceleration is the same as that of the net force

Newton’s Second Law of Motion

When a net force, F, acts on a mass, m it causes an acceleration, a.
Force = Mass x Acceleration F = ma

Relationship between a & F 
a
 F The acceleration, a, is directly proportional to the applied force, F.

Relationship between a and m
m
a
1

 The acceleration of an object is inversely  proportional to the mass, 

 Experiment to Find The Relationship between Force, Mass & Acceleration
  Relationship between
a & F a & m
Situation
 Both men are pushing the same mass but man A puts greater effort. So he moves faster.
 Both men exerted the same strength. But man B moves faster than man A.
Inference The acceleration produced by an object depends on the net force applied to it. 
The acceleration produced by an object depends on the mass 
Hypothesis The acceleration of the object increases when the force applied increases 
The acceleration of the object decreases when the mass of the object increases
Variables:
 Manipulated : Responding  : Constant       :
 Force Acceleration Mass
 Mass Acceleration Force 
Apparatus and Material
Ticker tape and elastic cords, ticker timer, trolleys, power supply and friction compensated runway and meter ruler.



An elastic cord
 is hooked over the trolley. The elastic cord is stretched until the end of the trolley. The trolley is pulled down the runway with the elastic cord being kept stretched by the same amount of force 
An elastic cord is hooked over a trolley
. The elastic cord is stretched until the end of the trolley. The trolley is pulled down the runway with the elastic cord being kept stretched by the same amount of force 


2.6 IMPULSE AND IMPULSIVE FORCE
  Impulse The change of momentum mv - mu Unit : kgms-1 or Ns
Impulsive Force
The rate of change of momentum in a collision or explosion
  Unit = N
m = mass u  = initial velocity v  = final velocity t   = time  
Longer period of time  
Impulsive force decrease
Effect of time
Impulsive force is inversely proportional to time of contact
Shorter period of time  
Impulsive force increase



   Situations for Reducing Impulsive Force in Sports 
Situations Explanation

Thick mattress with soft surfaces are used in events such as high jump so that the time interval of impact on landing is extended, thus reducing the impulsive force.  This can prevent injuries to the participants.

Goal keepers will wear gloves to increase the collision time.  This will reduce the impulsive force.

A high jumper will bend his legs upon landing. This is to increase the time of impact in order to reduce the impulsive force acting on his legs. This will reduce the chance of getting serious injury.

A baseball player must catch the ball in the direction of the motion of the ball. Moving his hand backwards when catching the ball prolongs the time for the momentum to change so as to reduce the impulsive force.
 Situation of Increasing Impulsive Force
Situations Explanation

A karate expert can break a thick wooden slab with  his bare hand that moves at a very fast speed. The short impact time results in a large impulsive force on the wooden slab. 

A massive hammer head moving at a fast speed is brought to rest upon hitting the nail.  The large change in momentum within a short time interval produces a large impulsive force which drives the nail into the wood.


A football must have enough air pressure in it so the contact time is short.  The impulsive force acted on the ball will be bigger and the ball will move faster and further.

Pestle and mortar are made of stone.  When a pestle is used to pound chilies the hard surfaces of both the pestle and mortar cause the pestle to be stopped in a very short time.  A large impulsive force is resulted and thus causes these spices to be crushed easily.
 Example 1 A 60 kg resident jumps from the first floor of a burning house.  His velocity just before landing on the ground is 6 ms
-1
. (a) Calculate the impulse when his legs hit the ground. (b) What is the impulsive force on the resident’s legs if he bends upon landing and takes 0.5 s to stop? (c) What is the impulsive force on the resident’s legs if he does not bend and stops in 0.05 s? (d) What is the advantage of bending his legs upon landing? 
 
Example 2 Rooney kicks a ball with a force of 1500 N.  The time of contact of his boot with the ball is 0.01 s.  What is the impulse delivered to the ball?  If the mass of the ball is 0.5 kg, what is the velocity of the ball?

 source:http://notaspmfizik.blogspot.com/2011/03/spm-f4-forces-motion.html




Analysing Momentum.

What is Momentum?
  1. It is more difficult for a large ship filled with cargo to dock in a harbour than a small ship.
  2. It is easier for a person to stop while he is walking than while he is running.
  3. the resistance to a change the state of motion of an object depends on two factors-the mass and velocity of the object.

Linear Momentum

  1. It is always harder to stop a massive object moving at high velocity.
  2. The above activity serves to explain a concept in physics called momentum.
  3. The linear momentum,p, of an object of mass,m, moving with a velocity,v, is defined as the product of its mass and velocity.
            MOMENTUM= MASS x VELOCITY
                                  p= mv

Conservation of Momentum
  1. Like energy, momentum is also a conserved quantity.
  2. The term conservation is used if the same amount of matter remains the same after an event occurs.
  3. The Principle of conservation of momentum states that:
The total momentum of a system is constant, if no external force acts on the system.

  1. An example of an external force is friction.
  2. To be precise, the Principle of conservation of momentum is true for an isolated system. An isolated system is one where the sum of external forces acting on the system is zero.
  3. An example of an external force is friction.
  4. To be precise, the Principle of conservation of momentum is true for an isolated system. An isolated system is one where the sum of external forces acting on the system is zero.
  5. The Principle shall be discussed in two situation.

The principle of conservation of linear momentum states that the total linear momentum of a closed system is constant or the linear momentum of a system before and after a collision is conserved if there is no external force acting on it.

Elastic collision.
  • Linear momentum, kinetic energy and total energy are conserved.
  • Objects do not stick together after collision.
    m1u1+m1v1=m2u2+m2v2.
Inelastic collision.
  • Only linear momentum and total energy are conserved and there is a loss in kinetic energy.
  • Objects are stuck together after collision.
   m1u1+m2v2=(m1+m2)v.
Explosion
- In an explosion where two objects move in opposite directions, the total linear momentum before and the explosion is conserved at zero.
Relationship between mass and inertia.

  1. A shopper in a supermarket observes that it is always easier to start moving an empty grocery cart than a full cart.
  2. Similarity, it is easier to stop an empty cart than a full one if both are moving at the same speed towards the shopper.
  3. Thus, the resistance to change to velocity (including zero velocity) i.e. the inertia is related to the mass of the object. Since the mass for the full cart is more than the empty one, and the full cart offers more inertia, it is obvious that inertia increases when mass increases.
  4. By the same reasoning, a bowling ball is harder to stop than a hollow rubber ball of the same size.
  5. However, inertia is a phenomenon. It has no unit even it is closely related to mass, which has a SI unit of kilogram.

Conclusionly, inertia of an object is the object's resistance to change in its motion or its state of rest. If mass increase, then inertia will increase.Magnitude of inertia (mass) can be shown by an inertia balance.