Work Incline Problem: Find Distance Up Incline for Block w/ No/Low Friction

[xjasonx1]

Homework Statement

A 230 g block is pressed against a spring of force constant 1.20 kN/m until the block compresses the spring 10.0 cm. The spring rests at the bottom of a ramp inclined at 60.0° to the horizontal. Use energy considerations, determine how far up the incline the block moves before it stops under the following conditions.
(a) if there is no friction between block and ramp
(b) if the coefficient of kinetic friction is 0.400

I am not sure if my assumption is correct. The block starts is released at rest so the in\Deltaitial velocity is zero and the question is asking for the distance at rest so the final velocity is zero making
$$\Delta$$K zero.

 

[tiny-tim]

I am not sure if my assumption is correct. The block starts is released at rest so the in\Deltaitial velocity is zero and the question is asking for the distance at rest so the final velocity is zero making
$$\Delta$$K zero.

Hi xjasonx1!

(have a delta: ∆ )

Yes, ∆KE is zero.

 

[kerimek]

is this correct?

Your assumption is correct. Since the block starts at rest, the initial kinetic energy is zero and the final kinetic energy is also zero as the block comes to a stop. Therefore, the change in kinetic energy, and thus the work done by friction, is also zero. This means that the total work done by the spring is equal to the work done by gravity in moving the block up the incline. Using the formula W = mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height, we can find the distance up the incline for both cases.

(a) If there is no friction, the work done by gravity is equal to the work done by the spring. We can set up the equation as follows:

W_spring = W_gravity
1/2kx^2 = mgh

Solving for h, we get h = 1/2kx^2/mg. Plugging in the given values, we get h = 1/2(1.20 kN/m)(0.10 m)^2/(0.230 kg)(9.8 m/s^2) = 0.002 m or 2 mm.

Therefore, the block will move up the incline by 2 mm before coming to a stop.

(b) If the coefficient of kinetic friction is 0.400, the work done by friction must be taken into account. We can set up the equation as follows:

W_spring = W_gravity + W_friction
1/2kx^2 = mgh + μmgd

Where μ is the coefficient of kinetic friction and d is the distance the block travels up the incline before coming to a stop.

Solving for d, we get d = (1/2kx^2 - μmg)/mg. Plugging in the given values, we get d = (1/2(1.20 kN/m)(0.10 m)^2 - 0.400(0.230 kg)(9.8 m/s^2))/(0.230 kg)(9.8 m/s^2) = 0.013 m or 13 mm.

Therefore, with a coefficient of kinetic friction of 0.400, the block will move up the incline by 13 mm before coming to a stop. This is significantly higher than the distance of 2 mm