How Does Sand Affect the Dynamics of a Conveyor Belt System?

[dkgojackets]

Homework Statement

5 part question here. "Sand from a stationary hopper falls on a moving conveyor belt at a rate of 5.81 kg/s. The belt is supported by frictionless rollers and moves at .645 m/s under the action of a horizontal external force supplied by the motor that drives the belt."

1. Find the sand's rate of change of momentum in the horizontal direction (answer in units of N)

2. Find the frictional force exerted by the belt on the sand (in units N)

3. Find the magnitude of the external force supplied by the motor

4. Find the work done by the external force each second

5. Find the kinetic energy acquired by the falling sand each second due to the change in its horizontal motion.

Homework Equations

I know I have to derivate momentum with respect to time, use conservation of momentum, and force equations. Work-energy theorem seems to be needed too.

The Attempt at a Solution

For one, would I just multiply the velocity of the sand on the belt by the changing mass? I would get v(dm/dt), which is a dP/dt. Seems too easy. I'm not quite sure how to get frictional force without the coefficient. It is acting against the external force in the positive horizontal direction on my free-body diagram, but I can't figure out how to solve it. I'm pretty lost on the next two, and especiallt the last one. There is no change in horizontal motion until the sand hits the belt, so why would it acquire KE while falling from it?

 

[berkeman]

You have the right idea -- just use the change in horizontal velocity and mass to give you all the answers. Assume that the sand does not fall very far, so its vertical motion does not enter into the calculations.

 

[dkgojackets]

I have one and five. I am not sure how to calculate the force of friction without the coefficient. I am guessing the magnitude of the external force would equal that since the sand is not accelerating on the conveyor?

 

[arildno]

You might have interest in the following tutorial:
https://www.physicsforums.com/showthread.php?t=72176

 

[dkgojackets]

Thank you, very helpful. I also needed the one with the rocket expelling fuel.

The only part I have left is 2, the frictional force exerted by the belt.

 

[dkgojackets]

still need help on frictional force please

 

[berkeman]

still need help on frictional force please

I think you must already have it. The only force on the sand is frictional force. The sand doesn't have to be slipping for you to consider the force's origin to be from friction.

 

[dkgojackets]

Yeah I got it. The first three parts of the question were the same value.

 

[OLeee]

Why is the work done twice the gain in KE of the sand