Physics Help PLEASE!!?

Question

I keep trying and trying but just cannot get the right answer!! Can someone please explain in simple terms how to find the answer for this problem-It should be 348 Newtons but that is NOT what i am getting!!


- A 30kg child is swinging at the end of a 5 meter long rope. The child is traveling at 3 m/s at the lowest point of its motion. Determine the tension in the rope at this point.

Answer 1

Yes, it's 348 Newtons --- at least if you take g = 9.80 m/s^2.

Here's how one gets this result. I'll carefully EXPLAIN every step, rather than just throw a final formula at you:

At the lowest point of the motion, the UPWARD acceleration due to the motion in a circular arc is given by

a = v^2 / r = (3^2) / 5 m/s^2 = 9 / 5 m/s^2.

In order to produce this acceleration, there needs to be a NET FORCE F upwards of

F = m a = 30 (9/5) kgm/s^2 = 270/5 N = 54 N.

(I'll BET that's the answer you got.) However, that is NOT the TENSION, T, in the ROPE! Why not? --- Because GRAVITY is also acting on the child, with a force of mg DOWNWARDS. That means that:

F = T - mg [see ### below] so that T = F+ mg = [54 + 30(9.80)]N

= [54 + 294] N = 348 N.

QED

Live long and prosper.

### P.S. It's easy to miss this step, that the weight of the child has also to be counteracted by the rope. But think of it this way: Even without swinging, the STATIC weight of the child already requires a tension in the rope of 30(9.8)N = 294N. So, to give the child the UPWARD acceleration implied by swinging requires a greater force than 294N --- greater by precisely the amount that will give the particular acceleration worked out from ' v ' and 'r.'

This problem also serves as a reminder of the general situation, that acceleration is caused by the NET force in some given direction; so alwys look around for other forces that may be present, not just the one in the right direction that first comes to mind!

Answer 2

F total = F centrifugal + F weight

= mv^2 / r + mg = 30 kg x (3 m/s)^2 / 5 m
+ 30 kg x 9.8 m/s^2

= 54 N + 294 N = 348 N.