Center of Mass and Momentum: Study Notes and Problem Set

Center of Mass and Momentum

Center of Mass

The center of mass (CM) of an object or system is the unique point at which the weighted relative positions of all mass elements are balanced in all directions. The position of the CM for a system of discrete particles is given by:

• $x_{CM} = \frac{\sum m_i x_i}{\sum m_i}$

• $y_{CM} = \frac{\sum m_i y_i}{\sum m_i}$

• $z_{CM} = \frac{\sum m_i z_i}{\sum m_i}$

For a continuous mass distribution, these formulas are replaced by integrals:

• $x_{CM} = \frac{1}{M} \int x \, dm$

• $y_{CM} = \frac{1}{M} \int y \, dm$

• $z_{CM} = \frac{1}{M} \int z \, dm$

The CM is crucial in stability analysis, rotation, and motion prediction.

Example: Three-Particle System

Find the CM of three particles located at specific positions:

• Use the discrete formula for each coordinate.

• Insert the given masses and positions to solve for $x_{CM}$, $y_{CM}$, $z_{CM}$.

Momentum

Momentum is a physical quantity that describes the motion of an object. It is defined as the product of an object's mass and its velocity:

• $\vec{p} = m\vec{v}$

Momentum is a vector quantity, meaning it has both magnitude and direction.

Example

A bullet traveling at 900 m/s has a momentum of 4.5 kg·m/s. What is its mass?

• Rearrange $p = mv$ to solve for $m$:

• $m = \frac{p}{v} = \frac{4.5}{900} = 5.0 \times 10^{-3}$ kg = 5.0 g

Change in Momentum

The change in momentum of an object occurs when its velocity changes. This change is expressed as:

• $\Delta \vec{p} = m\Delta \vec{v}$

Newton's second law can be expressed in terms of momentum as follows:

• $\vec{F}_{net} = \frac{d\vec{p}}{dt}$

Example

A 10 kg roller skater walks at 1 m/s, then bounces back in the opposite direction at 4 m/s. What is the change in momentum?

• Calculate $\Delta \vec{p} = m(v_{final} - v_{initial}) = 10 \times (-4 - 1) = -50$ kg·m/s

• The negative sign indicates the direction of the change.

Impulse

Impulse is the measure of the effect of a force acting over a time interval. It is defined as:

• $\vec{J} = \vec{F} \Delta t$

• Impulse is also equal to the change in momentum: $\vec{J} = \Delta \vec{p}$

Example

If a bat strikes a ball, the impulse delivered during the short contact time changes the ball's momentum.

Collisions

Collisions between objects are categorized based on whether kinetic energy is conserved.

1. Elastic Collision

• Both momentum and kinetic energy are conserved.

• Objects bounce off each other without lasting deformation or heat.

• Advanced: Glancing collisions require 2D treatment.

Example: Head-on collision of two billiard balls:

• Final velocities: $v_{1f} = \frac{m_1 - m_2}{m_1 + m_2} v_{1i} + \frac{2m_2}{m_1 + m_2} v_{2i}$

• $v_{2f} = \frac{2m_1}{m_1 + m_2} v_{1i} + \frac{m_2 - m_1}{m_1 + m_2} v_{2i}$

2. Inelastic Collision

• Momentum is conserved, but kinetic energy is not.

• Some kinetic energy is transformed into other forms (heat, deformation).

Example: A car crash where vehicles crumple and lose some kinetic energy.

3. Perfectly/Completely Inelastic Collision

• A special case of inelastic collision where the colliding objects stick together and move as one mass after collision.

• Momentum is conserved, but kinetic energy is partially lost.

Example: Two clay balls colliding and sticking together upon impact.

Problem Set

1. An 1100 kg car moving at 21 m/s takes 0.22 s to stop when it hits a dirt bank. What is the average force exerted by the dirt on the car?

2. Given a 20 g bullet fired at 320 m/s from a 4.5 kg gun, (a) calculate the impulse acting on the gun. (b) Calculate the final speed of the gun. (c) The person's hand moves back 0.025 s when the gun is fired. Calculate the average force exerted by the gun.

3. A 0.5 kg hammer moving at 3.5 m/s hits a 5.0 kg mass at rest. If the hammer and mass stick together, what will be their final velocity (magnitude and direction)?

4. A 60-kg ice skater initially at rest on the track of 7.0 m/s, suddenly a 2.0 × 103 kg load of coal is dropped vertically onto the skater. What’s the skater’s velocity after the impact, assuming no friction?

5. A baseball bat of mass 0.9 kg is swung at velocity 40 m/s and hits a 0.15 kg ball at rest. The ball rebounds at 48 m/s. The contact time is 0.01 s. Find the impulse delivered to the ball and the average force exerted by the bat.

6. A 40 g bullet is fired at 900 m/s into a 5.0 kg wood block at rest, and they stick together. Find the final velocity of the system.

7. A 90 kg football player runs north at 8.0 m/s and collides with a 95 kg football player from 3.5 m/s toward the south. If they stick together after the collision, what is their final velocity?

8. In a head-on collision, a 1.5 kg cart moving at 2.0 m/s collides with a 2.0 kg cart moving at 1.0 m/s in the opposite direction. If the collision is elastic, calculate the final velocities of both carts.

9. Two cars, one of mass 1200 kg and the other 1500 kg, are traveling at 20 m/s and 15 m/s, respectively, in the same direction. If they collide and stick together, what is their common velocity after the collision?

10. A 0.02 kg bullet is fired at 400 m/s into a 1.0 kg block at rest on a frictionless surface. After the collision, the block and bullet move together at 3.0 m/s. (a) Calculate the change in kinetic energy. (b) What fraction of the bullet’s kinetic energy is transferred to the pot?