Sand falls vertically on a conveyor belt at a rate of 0.1 kg/s. In order to keep the belt moving at a uniform speed of 2 m/s, the force required to be applied on the belt is :
- (a)0 N
- (b)0.2 N
- (c)1.0 N
- (d)2.0 N
Correct — B, 0.2 N. The belt must give each kilogram of sand the belt's speed. The force equals the rate of change of momentum: F = v x (dm/dt) = 2 m/s x 0.1 kg/s = 0.2 N. (The sand lands at zero horizontal speed and must be accelerated to 2 m/s.)
- (a)0 N — Zero would apply only if no new mass were being added; here sand is continuously loaded and must be sped up, so a force is needed.
- (c)1.0 N — This does not match F = v(dm/dt) = 2 x 0.1 = 0.2 N; it overestimates the force fivefold.
- (d)2.0 N — This equals v x 1 kg/s, i.e. it wrongly uses a mass rate of 1 kg/s instead of the given 0.1 kg/s.
When mass is added to a moving system, Newton's second law in its general form F = d(mv)/dt applies. For a belt at constant speed v with sand piling on at rate dm/dt, the belt's speed is unchanged, so the force needed is F = v (dm/dt) — the rate at which momentum must be supplied to the incoming sand.
The belt speed is constant, so there is no v-changing term; the only momentum change is from new sand acquiring speed v. Plugging v = 2 m/s and dm/dt = 0.1 kg/s gives 0.2 N. The '2.0 N' trap uses the wrong mass rate.
- General form of Newton's second law: F = d(mv)/dt.
- For mass added at constant belt speed: F = v (dm/dt).
- Here F = 2 m/s x 0.1 kg/s = 0.2 N.
- The falling sand starts with zero horizontal velocity and is accelerated to the belt speed.
- Using the wrong mass rate (1 kg/s instead of 0.1 kg/s).
- Thinking no force is needed because the speed is constant — new mass must still be accelerated.
Asked as the force to keep a belt moving while mass is loaded, testing F = v(dm/dt).
No directly related past PYQ was found.
- practice — not a real PYQ
Sand falls on a belt moving at 3 m/s at 0.2 kg/s. The force needed to maintain the speed is
- (a)0.2 N
- (b)0.6 N
- (c)1.5 N
- (d)0 N
Answer(b) 0.6 N (F = 3 x 0.2).
- practice — not a real PYQ
The general form of Newton's second law is
- (a)F = ma only
- (b)F = d(mv)/dt
- (c)F = mv
- (d)F = v/t
Answer(b) F = d(mv)/dt.