Motion of the Center of Mass – Physics Problem
Question:
Consider two particles of equal mass m. The forces on the particles are:
- 𝐅₁ = 0 (no force on the first particle)
- 𝐅₂ = 𝐅₀
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 = F₁ + F₂ = 0 + F₀ = F₀
acm = F₀ / 2m
Acceleration of CM: acm(t) = (F₀ / 2m) î
Step 2: Velocity of the Center of Mass
Integrate acceleration with respect to time:
vcm(t) = ∫ acm(t) dt = ∫ (F₀ / 2m) dt = (F₀ / 2m)t
Velocity of CM: vcm(t) = (F₀ / 2m)t î
Step 3: Position of the Center of Mass
Integrate velocity with respect to time:
xcm(t) = ∫ vcm(t) dt = ∫ (F₀ / 2m)t dt = (F₀ / 4m)t²
Position of CM: xcm(t) = (F₀ / 4m)t² î
Summary:
- Acceleration: (F₀ / 2m) î
- Velocity: (F₀ / 2m)t î
- Position: (F₀ / 4m)t² î