Circular motion and work done by non conservative forces

[henry3369]

Homework Statement

A ball of mass m is attached to a string of length L. It is being swung in a vertical circle with enough speed so that the string remains taut throughout the ball's motion.(Figure 1) Assume that the ball travels freely in this vertical circle with negligible loss of total mechanical energy. At the top and bottom of the vertical circle, the ball's speeds are v_t and v_b, and the corresponding tensions in the string are T⃗_t and T⃗_b. T⃗_t and T⃗_b have magnitudes T_t and T_b.

Express the difference in tension in terms of m and g. The quantities v_t and v_b should not appear in your final answer.

Homework Equations

K_i+U_i+W_NC = K_f+U_f

The Attempt at a Solution

K_i+W_NC = K_f+U_f
W_NC = K_f+U_f - K_i
W_NC = (1/2)mv_t² + mg2L - (1/2)mv_b²Top:
T_t = (mv_t²/L) - mg
Bottom:
T_b = mg - (mv_b²/L)

I don't know what to do after this. I have work done by non conservative forces, but I'm not sure how to relate this to the difference in tensions.

 

[Orodruin]

There is no non-conservative force in this problem. As stated explicitly:

Assume that the ball travels freely in this vertical circle with negligible loss of total mechanical energy.

You should be able to use conservation of total energy (including the gravitational potential energy) in order to relate the velocities, and thus also the tensions.

 

[henry3369]

There is no non-conservative force in this problem. As stated explicitly:You should be able to use conservation of total energy (including the gravitational potential energy) in order to relate the velocities, and thus also the tensions.

Oh okay. I got the answer when the work done by non-conservative force = 0. If the problem didn't explicitly state that no mechanical energy is lost, could you still assume that W_NC = 0?

 

[Orodruin]

No, you would have to account for the additional work done on the system.

 

[Maged Saeed]

If the problem didn't explicitly state that no mechanical energy is lost, could you still assume that WNC = 0?

Yes, Because the loss in mechanical energy is equal to the work done by the non-conservative forces in an isolated systems