Motion of the Center of Mass β Physics Problem
Question:
Consider two particles of equal mass m. The forces on the particles are:
- π β = 0 (no force on the first particle)
- π
β = π
ββ―
If both particles are initially at rest at the origin, determine the: acceleration, velocity, and position of the center of mass as a function of time.
Answer with Full Step-by-Step Explanation:
Step 1: Acceleration of the Center of Mass
The acceleration of the center of mass is given by:
acm = Fnet / Mtotal
Total mass: Mtotal = m + m = 2m
Net force: Fnet = Fβ + Fβ = 0 + Fββ― = Fββ―
acm = Fβ / 2mβ―
Acceleration of CM: acm(t) = (Fβ / 2m)β―iΜ
Step 2: Velocity of the Center of Mass
Integrate acceleration with respect to time:
vcm(t) = β« acm(t) dt = β« (Fβ / 2m) dt = (Fβ / 2m)tβ―
Velocity of CM: vcm(t) = (Fβ / 2m)tβ―iΜ
Step 3: Position of the Center of Mass
Integrate velocity with respect to time:
xcm(t) = β« vcm(t) dt = β« (Fβ / 2m)t dt = (Fβ / 4m)tΒ²β―
Position of CM: xcm(t) = (Fβ / 4m)tΒ²β―iΜ
Summary:
- Acceleration: (Fβ / 2m)β―iΜ
- Velocity: (Fβ / 2m)tβ―iΜ
- Position: (Fβ / 4m)tΒ²β―iΜ