Conservation of Momentum, Momentum | Physics

Learning objectives

Section Key Terms

angular momentum

isolated system

law of conservation of momentum

Conservation of Momentum

It is important we realize that momentum is conserved during collisions, explosions, and other events involving objects in motion. To say that a quantity is conserved means that it is constant throughout the event. In the case of conservation of momentum, the total momentum in the system remains the same before and after the collision.

You may have noticed that momentum was not conserved in some of the examples previously presented in this chapter. where forces acting on the objects produced large changes in momentum. Why is this? The systems of interest considered in those problems were not inclusive enough. If the systems were expanded to include more objects, then momentum would in fact be conserved in those sample problems. It is always possible to find a larger system where momentum is conserved, even though momentum changes for individual objects within the system.

For example, if a football player runs into the goalpost in the end zone, a force will cause him to bounce backward. His momentum is obviously greatly changed, and considering only the football player, we would find that momentum is not conserved. However, the system can be expanded to contain the entire Earth. Surprisingly, Earth also recoils—conserving momentum—because of the force applied to it through the goalpost. The effect on Earth is not noticeable because it is so much more massive than the player, but the effect is real.

Next, consider what happens if the masses of two colliding objects are more similar than the masses of a football player and Earth—in the example shown in Figure_08_02_collision of one car bumping into another. Both cars are coasting in the same direction when the lead car, labeled m₂, is bumped by the trailing car, labeled m₁. The only unbalanced force on each car is the force of the collision, assuming that the effects due to friction are negligible. Car m1 slows down as a result of the collision, losing some momentum, while car m2 speeds up and gains some momentum. If we choose the system to include both cars and assume that friction is negligible, then the momentum of the two-car system should remain constant. Now we will prove that the total momentum of the two-car system does in fact remain constant, and is therefore conserved.

View original figure (external; image not reproduced in this text edition)

Original figure description: Revise alt text to: An illustration shows before and after diagrams of two cars, one in front of the other. The car in the back is labeled m one and the car in the front is labeled m two. Both diagrams are labeled System of Interest. In the before diagram, a velocity vector points from the car in the back to the car in the front. A second velocity vector of equal magnitude points from the car in the front to the right. Two equations are shown: p one plus p two equals p total and net F equals zero. In the after diagram, the velocity vector pointing from the car in the back is shorter. The velocity vector pointing from the car in the front is longer. They are labeled v one prime and v two prime. The equation p one prime plus p two prime equals p total is shown.

Car of mass m₁ moving with a velocity of v₁ bumps into another car of mass m₂ and velocity v₂. As a result, the first car slows down to a velocity of v′₁ and the second speeds up to a velocity of v′₂. The momentum of each car is changed, but the total momentum p_tot of the two cars is the same before and after the collision if you assume friction is negligible.

Using the impulse-momentum theorem, the change in momentum of car 1 is given by

Δ p_{1} = F_{1} Δ t,

where F₁ is the force on car 1 due to car 2, and $Δ t$ is the time the force acts, or the duration of the collision.

Similarly, the change in momentum of car 2 is $Δ p_{2} = F_{2} Δ t$ where F₂ is the force on car 2 due to car 1, and we assume the duration of the collision $Δ t$ is the same for both cars. We know from Newton’s third law of motion that F₂ = –F₁, and so $Δ p_{2} = - F_{1} Δ t = - Δ p_{1}$ .

Therefore, the changes in momentum are equal and opposite, and $Δ p_{1} + Δ p_{2} = 0$ .

Because the changes in momentum add to zero, the total momentum of the two-car system is constant. That is,

p_{1} + p_{2} = constant

p_{1} + p_{2} = {p^{'}}_{1} + {p^{'}}_{2},

where p′₁ and p′₂ are the momenta of cars 1 and 2 after the collision.

This result that momentum is conserved is true not only for this example involving the two cars, but for any system where the net external force is zero, which is known as an isolated system. The law of conservation of momentum states that for an isolated system with any number of objects in it, the total momentum is conserved. In equation form, the law of conservation of momentum for an isolated system is written as

p_{tot} = constant

p_{tot} = {p^{'}}_{tot},

where p_tot is the total momentum, or the sum of the momenta of the individual objects in the system at a given time, and p′_tot is the total momentum some time later.

The conservation of momentum principle can be applied to systems as diverse as a comet striking the Earth or a gas containing huge numbers of atoms and molecules. Conservation of momentum appears to be violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force. For example, in the collision of two cars considered above, the two-car system conserves momentum while each one-car system does not.

Angular Momentum in Figure Skating

So far we have covered linear momentum, which describes the inertia of objects traveling in a straight line. But we know that many objects in nature have a curved or circular path. Just as linear motion has linear momentum to describe its tendency to move forward, circular motion has the equivalent angular momentum to describe how rotational motion is carried forward.

This is similar to how torque is analogous to force, angular acceleration is analogous to translational acceleration, and mr² is analogous to mass or inertia. You may recall learning that the quantity mr² is called the rotational inertia or moment of inertia of a point mass m at a distance r from the center of rotation.

We already know the equation for linear momentum, p = mv. Since angular momentum is analogous to linear momentum, the moment of inertia (I) is analogous to mass, and angular velocity $(ω)$ is analogous to linear velocity, it makes sense that angular momentum (L) is defined as

L = I ω

Angular momentum is conserved when the net external torque ( $τ$ ) is zero, just as linear momentum is conserved when the net external force is zero.

Figure skaters take advantage of the conservation of angular momentum, likely without even realizing it. In Figure_08_02_iceskater, a figure skater is executing a spin. The net torque on her is very close to zero, because there is relatively little friction between her skates and the ice, and because the friction is exerted very close to the pivot point. Both F and r are small, and so $τ$ is negligibly small.

View original figure (external; image not reproduced in this text edition)

Original figure description: An illustration shows an ice skater in two positions. She is spinning on the tip of her skate with her arms extended in the first position. In the second position, her arms are pulled in as she is spinning. In the first position, an arrow labeled w shows the direction of her spin and the equation L equals I times w is shown. In the second position, an arrow labeled w prime shows the direction of her spin and the equation L equals I prime times w prime is shown.

(a) An ice skater is spinning on the tip of her skate with her arms extended. In the next image, (b), her rate of spin increases greatly when she pulls in her arms.

Consequently, she can spin for quite some time. She can do something else, too. She can increase her rate of spin by pulling her arms and legs in. Why does pulling her arms and legs in increase her rate of spin? The answer is that her angular momentum is constant, so that L = L′.

Expressing this equation in terms of the moment of inertia,

I ω = I^{'} ω^{'},

where the primed quantities refer to conditions after she has pulled in her arms and reduced her moment of inertia. Because I′ is smaller, the angular velocity $ω^{'}$ must increase to keep the angular momentum constant. This allows her to spin much faster without exerting any extra torque.

A video is also available that shows a real figure skater executing a spin. It discusses the physics of spins in figure skating.

Based on the equation L = Iω, how would you expect the moment of inertia of an object to affect angular momentum? How would angular velocity affect angular momentum?

Large moment of inertia implies large angular momentum, and large angular velocity implies large angular momentum.
Large moment of inertia implies small angular momentum, and large angular velocity implies small angular momentum.
Large moment of inertia implies large angular momentum, and large angular velocity implies small angular momentum.
Large moment of inertia implies small angular momentum, and large angular velocity implies large angular momentum.

Check Your Understanding

When is momentum said to be conserved?

When momentum is changing during an event
When momentum is increasing during an event
When momentum is decreasing during an event
When momentum is constant throughout an event

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Section Summary

The law of conservation of momentum is written p_tot = constant or p_tot = p′_tot (isolated system), where p_tot is the initial total momentum and p′_tot is the total momentum some time later.
In an isolated system, the net external force is zero.
Conservation of momentum applies only when the net external force is zero, within the defined system.

Key Equations

law of conservation of momentum	p_tot = constant, or p_tot = p′_tot
conservation of momentum for two objects	p₁ + p₂ = constant, or p₁ + p₂ = p′₁ + p′₂
angular momentum	L = I $ω$

Concept Items

What is angular momentum?

The sum of moment of inertia and angular velocity
The ratio of moment of inertia to angular velocity
The product of moment of inertia and angular velocity
Half the product of moment of inertia and square of angular velocity

What is an isolated system?

A system in which the net internal force is zero
A system in which the net external force is zero
A system in which the net internal force is a nonzero constant
A system in which the net external force is a nonzero constant

Critical Thinking

A billiards ball rolling on the table has momentum p₁. It hits another stationary ball, which then starts rolling. Considering friction to be negligible, what will happen to the momentum of the first ball?

It will decrease.
It will increase.
It will become zero.
It will remain the same.

A ball rolling on the floor with momentum p₁ collides with a stationary ball and sets it in motion. The momentum of the first ball becomes p'₁, and that of the second becomes p'₂. The directions of p₁, p'₁, and p'₂ are all the same. Compare the magnitudes of p₁ and p'₂.

Momenta p₁ and p'₂ are the same in magnitude.
The sum of the magnitudes of p₁ and p'₂ is zero.
The magnitude of p₁ is greater than that of p'₂.
The magnitude of p'₂ is greater than that of p₁.

Two cars are moving in the same direction. One car with momentum p₁ collides with another, which has momentum p₂. Their momenta become p'₁ and p'₂ respectively. Considering frictional losses, compare (p'₁ + p'₂ ) with (p₁ + p₂).

The value of (p'₁ + p'₂ ) is zero.
The values of (p₁ + p₂) and (p'₁ + p'₂ ) are equal.
The value of (p₁ + p₂) will be greater than (p'₁ + p'₂ ).
The value of (p'₁ + p'₂ ) will be greater than (p₁ + p₂).

Test Prep Multiple Choice

In the equation L = Iω, what is I?

Linear momentum
Angular momentum
Torque
Moment of inertia

Give an example of an isolated system.

A cyclist moving along a rough road
A figure skater gliding in a straight line on an ice rink
A baseball player hitting a home run
A man drawing water from a well

Test Prep Short Answer

Under what condition(s) is the angular momentum of a system conserved?

When net torque is zero
When net torque is not zero
When moment of inertia is constant
When both moment of inertia and angular momentum are constant

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Test Prep Extended Response

Why does a figure skater spin faster if he pulls his arms and legs in?

Due to an increase in moment of inertia
Due to an increase in angular momentum
Due to conservation of linear momentum
Due to conservation of angular momentum

Glossary

angular momentum: the product of the moment of inertia and angular velocity
isolated system: system in which the net external force is zero
law of conservation of momentum: when the net external force is zero, the total momentum of the system is conserved or constant

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