Please help: A box on a frictionless plane, find the power based on work and

[nickclarson]

[SOLVED] Please help: A box on a frictionless plane, find the power based on work and

Homework Statement

A box of mass M = 7.68 kg is at rest at the bottom of a frictionless inclined plane.


The box is attached to a string that pulls with a constant tension T = 113 N. Write the general equation that gives the work done by the tension when the box has moved a distance x along the plane. Now, find the speed of the box as a function of x and the angle of the plane. Finally, for a distance of x = 6.38 m and an angle of 30.5°, determine the power delivered by the tension in the string as a function of x.

Homework Equations

$$K=\frac{1}{2}mv^{2}$$
$$P=FV$$

The Attempt at a Solution

I know I am supposed to multiply the net force by velocity for the last part, but I need to find the velocity.

I figured I could use kinetic energy to find the velocity using the ke work theorem. So...

$$KE_{f}-KE_{i}=W$$

And since the initial energy is 0

$$KE_{f}=W$$

Am I on the right track?

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Edit: Solved

Here's the final equation I derived:

$$P=T\sqrt{\frac{2x(T-mgsin\theta)}{m}}$$

 

[Nate810]

For x = 6.38 m and an angle of 30.5°, the power delivered by the tension in the string is P = 544.2 W.

 

[m2003]

sin\theta

I would first commend the student for their attempt at solving the problem and for seeking help when they were unsure. I would also clarify that the work done by the tension is not equal to the kinetic energy, but rather the change in kinetic energy. So the equation should be KE_f - KE_i = W, as the student correctly stated later in their attempt.

To find the velocity, I would suggest using the formula for work done by a constant force, W = Fd = mad, where d is the distance moved and a is the acceleration. Since the box is on a frictionless plane, the only force acting on it is the tension force, so we can rewrite this as W = Td = mad. Solving for a, we get a = T/m. We can then use the formula for constant acceleration, v^2 = v_0^2 + 2ad, where v_0 is the initial velocity (0 in this case) and d is the distance moved. Rearranging for v, we get v = sqrt(2ad). Plugging in the values for T and m, we get v = sqrt(2Tx/m).

To find the power, we can use the formula P = W/t, where W is the work done and t is the time taken. Since we are given a distance and an angle, we can use trigonometry to find the displacement in the x-direction, which is equal to the distance the box has moved. So we can rewrite the formula as P = (W/d)(d/t), where d is the distance moved and t is the time taken. We can then substitute in the formula for work done by the tension, W = Td, and the formula for velocity we found earlier, v = sqrt(2Tx/m). This gives us P = (Td/d)(d/v) = Tsqrt(2Tx/m). Finally, we can substitute in the values for T, x, and m to get the final equation for power, which the student correctly derived in their attempt.