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Momentum is the product of mass and velocity.

where p = momentum (kgm/s), m = mass (kg), and v = velocity (m/s)

Momentum is a vector quantity and "points" in the same direction as the velocity.

Stationary objects do not have momentum whereas moving objects do have momentum. Two identical objects have different momenta if their velocities are different. Two different objects with quite different masses can have the same momentum if the heavy object moves slow enough and the light object moves fast enough.

In order to change the momentum of an object, its velocity or mass must be changed. Most of the time it is the velocity that is changed. We can define the change in momentum, Dp or impulse as the product of the mass and the change in velocity.

• Dp = mDv

Remember, when an object's velocity is changing, it is accelerating and there must be a net force acting on it. We can discover the relationship between force and change in momentum by substituting at for Dv (a = Dv/Dt)

• Dp = maDt

If F = ma, then

• Dp = FDt

Recall the egg demonstration. We could throw the raw egg into the sheet as hard as we could but we couldn't get it to break, but when it went into the wall, it broke apart like we all expected it to. Why?

Both eggs experienced the same change in momentum, but the way they were stopped was quite different. In the case of the wall, the time over which the force was applied was quite small so the force had to be very large.

Dp = FDt

In the case of the sheet, the time over which the force was exerted was much larger which allowed the force to be smaller, keeping the egg intact.

Dp = FDt

Momentum Transfer in Collisions

We observed two cases where some momentum was transferred in a collision.

A moving cart careened into the back of a stationary cart. It started out with a large momentum due to its large velocity. After it contacted and stuck to the 2nd cart, it was slowed down significantly. It had lost some momentum , which had been transferred to the 2nd cart. When we compared the momentum of the two carts after the collision with the momentum of the first cart before the collision, they were about equal. This means that momentum was conserved in the collision.

Case #2 Marbles and Collision in 2-D Experiment

We rolled a metal marble into a glass marble so that the collision was not straight on but at an angle. We determined the total momentum of the metal ball before the collision and compared it to the vector sum of the momenta of each sphere after the collision. We found that they were nearly equal.

From thest two experiments we discovered  the Law of Conservation of Momentum.

Law of Conservation of Momentum - In any collision, the total momentum after the collision equals the total momentum before the collision

Torque and Angular Motion

Just as an unbalanced force causes an translational acceleration, a torque causes an angular acceleration. If a force is applied through the center of mass of an object it will experience an translational acceleration but not an angular acceleration. Forces applied so that they do not pass through the center of mass will cause a change in angular velocity and result in angular acceleration. We define Torque, T as the cross product of the radius (distance of force from the center) and the applied force.

t = r x F

The units of torque are N*m

If the force is not perpendicular to the radius, we modify the above equation to

t = r x Fsinq

Torque is a vector quantity and we apply the right hand rule to determine the direction.

We can maximize the torque by

1) exerting a larger force
2) maximizing the radius
3) exerting the force perpendicular to the radius

An unbalanced force produces an acceleration. An unbalanced torque will produce an angular acceleration. Newton's 2nd Law equation applied to angular motion is

t = Ia

where 

• t = Torque (N*m)     
• a = angular acceleration (rad/s²)             
• I = moment of inertia (kg*m²)

The moment of inertia depends on both the mass of the object and the way the mass is distributed. The equation for spinning objects vary, but all have the mass multiplied by the square of the radius. When the mass is more spread out r is larger yielding a larger moment of inertia.

Angular Momentum

Just as objects moving in a straight line have momentum, objects that have angular velocity also have angular momentum. The equation for angular momentum is

L = Iw

where

• L = angular momentum           
• I = moment of inertia         
• w= angular velocity

Angular momentum is also a vector quantity. Its direction is determined by using the right hand rule.

While changing linear momentum requires an impulse Dp = FDt changing angular momentum requires an impulse tDt . And just with linear momentum, angular momentum is conserved.

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