Question

hopper drops gravel at a rate of 75 kg/s onto a conveyor belt that moves at a constant speed v = 2.2 m/s. (a) Determine the force needed to keep the conveyor belt moving. (b) What is the rate at which this force does work?​

Answer 1

Given:

A hopper drops gravel at a rate of 75 kg/s onto a conveyor belt which moves at a constant speed of v = 2.2 m/s

To find:

• Force needed to keep the conveyor belt moving
• Rate of work done by the force

Calculation:

Force is define does the instantaneous rate of change of momentum with respect to time. Since Momentum itself consists of of mass and velocity we can represent force in these two terms as well.

Let force be F and momentum be P ;

$\therefore \: F = \dfrac{dP}{dt}$

$= > \: F = \dfrac{d(mv)}{dt}$

Applying Product Rule Of Differentiation:

$= > \: F = m \dfrac{d(v)}{dt} + v \dfrac{d(m)}{dt}$

Since conveyor belt moves at constant speed , then dv/dt is zero.

$= > \: F = (m \times 0) + v \dfrac{d(m)}{dt}$

$= > \: F = v \times \dfrac{d(m)}{dt}$

$= > \: F = 2.2 \times 75$

$= > \: F = 165 \: N$

Now , Power can be calculated as the product of of force experienced and the velocity of the conveyor belt.

$\therefore \: power = force \times velocity$

$= > power = 165 \times 2.2$

$= > power = 363 \: watt$

So final answer:

• Force is 165 N
• Power is 363 Watt