Physics Center of Mass Quiz

Study Notes

The center of mass (Com) is the average position of all the mass points in a system.
To find the Com, use the formula: Kcom = (m1r1 + m2r2) / (m1+m2), where m1 and m2 are the masses and r1 and r2 are the positions of the masses.
The Com can be found by taking moments of the masses along a line.
The Com can also be calculated continuously by integrating over the mass distribution.

The position of the Com depends on your choice of origin.
(xcm)1 = m2R / (m1+m2) and (xcm)2 = m1R / (m1+m2), where R is the distance between the two masses.

The change in the position of the Com can be calculated using the formula: drcm = (m1dr1 + m2*dr2) / (m1+m2), where dr1 and dr2 are the changes in position of the masses.
For no movement of the Com, m1dr1 = -m2dr2.

The acceleration of the Com is given by acm = (m1a1 + m2a2) / (m1+m2).
In a pulley block system, the acceleration of the Com is: acm = -(m2-m1)^2 / (m1+m2)*g.

The momentum of a body is defined as P = mv, where m is the mass and v is the velocity.
Newton's second law states: F = dP / dt.
If the net external force on a system is zero, the momentum of the system is conserved.
This means that the velocity of the Com is constant, and the acceleration of the Com is zero.
The velocity of the Com can be zero, but this does not necessarily mean that the acceleration of the Com is zero.
For a system, the external force is equal to the rate of change of the momentum of the Com: Fext = dPcm / dt.

An explosion is a process where internal forces cause a rapid increase in kinetic energy.
In an explosion, the net external force on the system is zero, so the momentum of the system is conserved.
The change in kinetic energy during an explosion is equal to the energy released in the explosion: Q = Kf - Ki = 1/2 * mv^2f + 1/2 * mV^2i, where Kf is the final kinetic energy and Ki is the initial kinetic energy.

A collision is an event where two or more objects come into contact.
The coefficient of restitution (e) is a measure of how much kinetic energy is lost during a collision.
e = separation / muom = (V1-V2) / (U2-U1)

In an elastic collision, kinetic energy is conserved: KEi = KEf.
The momentum of the system is also conserved: Pi = Pf.
The final velocities of the objects can be calculated using the following equations:
- V1 = (m1-m2 / m1+m2) * U1 + 2m2 / m1+m2 * U2
- V2 = 2m1 / m1+m2 * U1 + (m2-m1 / m1+m2) * U2
The change in kinetic energy in an elastic collision is zero: ΔKE = 0.

When the two masses are equal, their velocities are interchanged after the collision.

In inelastic collisions, kinetic energy is not conserved: KEi ≠ KEf.
Momentum is still conserved: Pi = Pf.
The final velocities of the objects can be calculated using the following equation:
- V1 = (m1-m2 / m1+m2) * U1 + b*U2 (m1+m2 / m1+m2)
The change in kinetic energy in an inelastic collision is given by: ΔKE =~ 1/2 * m1*m2 / (m1+m2) * (V^2i - V^2f)

In a perfectly inelastic collision, the two objects stick together after the collision.
Momentum is still conserved: Pi = Pf.
Kinetic energy is not conserved: KEi ≠ KEf.
The final velocity of the two objects is given by: Vo comunes = (m1*U1 + m2 *U2) / (m1+m2).
The change in kinetic energy is given by: ΔKE = 1/2 * (m1 * m2 / (m1+m2)) *(V0)^2

In an oblique collision, the objects collide at an angle.
We use the following steps to analyze oblique collisions:
- Resolve the velocities of the objects into components perpendicular and parallel to the line of impact.
- Apply the conservation of momentum along both the parallel and perpendicular directions.
- If the collision is inelastic, use the coefficient of restitution to calculate the final velocities.
- If the collision is elastic, use the principle of kinetic energy conservation in addition to the momentum conservation equations.