Solve Physics Problem: Block Moving Up a Vertical Track

[KD-jay]

Homework Statement

A block of mass 0.640 kg is pushed against a horizontal spring of negligible mass until the spring is compressed a distance x. The force constant of the spring is 450 N/m. When it is released, the block travels along a frictionless, horizontal surface to point B, the bottom of a vertical circular track of radius R = 1.00 m, and continues to move up the track. The speed of the block at the bottom of the track is vB = 14.0 m/s, and the block experiences an average frictional force of 7.00 N while sliding up the track.
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(a) What is x?
(b) What speed do you predict for the block at the top of the track?

I've gotten (a) but can not figure out (b).

Homework Equations

Σ_nc=ΔKE + ΔU_g + ΔU_s

The Attempt at a Solution

(a)What is x?
Σ_nc=ΔKE + ΔU_g + ΔU_s
Σ_nc=(1/2)*m*v_f^2 - (1/2)*m*v_i^2 + mgh_f - mgh_i + (1/2)*k*x_f^2 - (1/2)*k*x_i^2
Σ_nc=(1/2)*m*v_f^2 - (1/2)*k*x_f^2
0=(1/2)*0.640*14^2 - (1/2)*450*x_f^2
x=0.528

This is the one I'm having trouble on
(b) What speed do you predict for the block at the top of the track?
I'm assuming that the work done by the friction is F*Δr = 7.00 * Π since the displacement is half the loop. I'm also assuming that the height at the top of the loop is 2 meters.

Σ_nc=ΔKE + ΔU_g + ΔU_s
Σ_nc=(1/2)*m*v_f^2 - (1/2)*m*v_i^2 + mgh_f - mgh_i + (1/2)*k*x_f^2 - (1/2)*k*x_i^2
Σ_nc=(1/2)*m*v_f^2 - mgh_f - (1/2)*k*x_f^2
-7.00*Π=(1/2)*0.640*v_f^2 - 0.640*9.81*2 - (1/2)*450*(0.528)^2
v_f=12.9 m/s

 

[Doc Al]

Looks like you messed up the sign of the gravitational PE in your last equation.

 

[thomate1]

I would like to commend the student for their efforts in solving this physics problem. It is clear that they have a good understanding of the relevant equations and have correctly calculated the value for x in part (a). However, I would like to offer some suggestions for part (b).

Firstly, it is important to note that the work done by friction is not simply F*Δr, as suggested by the student. The work done by friction is equal to the product of the frictional force and the displacement along the direction of the force. In this case, the displacement is not simply Π, but rather the length of the track that the block travels along.

Secondly, the student has correctly identified that the height at the top of the loop is 2 meters, but they have not taken into account the change in potential energy due to the spring. Remember, the spring is still compressed at the top of the loop, so it will still have some potential energy.

Finally, it is important to remember that energy is conserved in this system, so the total energy at the bottom of the track (kinetic energy + potential energy) will be equal to the total energy at the top of the track. Using this information and the correct equations, the student should be able to solve for the final speed at the top of the track.

Overall, the student has shown a good understanding of the problem and has made a good attempt at solving it. With a few adjustments and taking into account all of the relevant factors, I am confident that the student will be able to accurately calculate the final speed at the top of the track. Keep up the good work!