Power needed to keep velocity of conveyor belt constant

[diablo2121]

Homework Statement

A conveyor belt travels at a constant velocity $$v$$ while sand is poured onto it at rate of $$\frac{dm}{dt}$$. Find the power needed to drive the conveyor belt.

Homework Equations

$$P = \frac{dW}{dt} = F*v$$
$$F = m*\frac{dv}{dt}$$

The Attempt at a Solution

I have the equations needed to solve the problem, but I'm confused as to how I can translate $$\frac{dm}{dt}$$ into the equations.

 

[tiny-tim]

Hi diablo2121!

Homework Equations

…
$$F = m*\frac{dv}{dt}$$

Noooo … $$F = \frac{d(mv)}{dt}$$

(this is the full version of Newton's second law … f = m dv/dt is only a special case, valid only when dm/dt = 0)

 

[baba89]

In this scenario, the sand being poured onto the conveyor belt can be considered as a continuous stream of particles with a certain mass flow rate, \frac{dm}{dt}. This means that for every unit of time, a certain amount of mass is being added to the conveyor belt.

To incorporate this into the equations, we can substitute the value of mass, m, with the mass flow rate, \frac{dm}{dt}. This will give us:

P = \frac{dW}{dt} = (\frac{dm}{dt})*v

Next, we can use the equation F = m*\frac{dv}{dt} to solve for \frac{dv}{dt}, which represents the acceleration of the conveyor belt.

Since the velocity is constant, we can assume that the acceleration is zero. This means that \frac{dv}{dt} = 0. Substituting this into the equation for force, we get:

F = m*0 = 0

Therefore, the power needed to drive the conveyor belt is also zero, as there is no force required to maintain a constant velocity.

In conclusion, the power needed to keep the velocity of the conveyor belt constant is zero, as long as the mass flow rate of the sand remains constant and there is no external force acting on the belt.

 

[thomate1]

To solve this problem, we can use the concept of conservation of energy. The power needed to keep the velocity of the conveyor belt constant is equal to the rate at which energy is being transferred to the belt. This can be represented by the equation P = dW/dt, where P is power, dW is the change in energy, and dt is the change in time.

In this case, the energy being transferred to the belt is due to the sand being poured onto it. We can represent the rate at which sand is being poured onto the belt as dm/dt, where m is the mass of sand and t is time. This can be translated into the equation F = m*dv/dt, where F is the force being applied to the belt, m is the mass of sand, and dv/dt is the rate of change of velocity.

Now, we can substitute this into the equation for power, P = F*v, to get P = (m*dv/dt)*v. This can be simplified to P = m*v*dv/dt.

Therefore, the power needed to keep the velocity of the conveyor belt constant is directly proportional to the mass of sand being poured onto it and the velocity of the belt. We can use this equation to calculate the power needed for any given mass and velocity of sand.