What Is the Minimum Distance Below a Pivot for a Pendulum String to Stay Taut?

[Raziel2701]

Homework Statement

A pendulum of length L is initially held horizontal, and is then released. The string runs into a peg a distance d below the pivot. What is the smallest value of d for which the string remains taught at all times?

Homework Equations

The Attempt at a Solution

I did a free body diagram for the mass when it's winding about the peg. I got $$T+mg=\frac{mv^2}{L}$$

Then I used conservation of energy:

$$\frac{1}{2}mv^2 +mg(L-d)=mgL$$

I solved for v^2 in the force equation and plugged into my conservation of energy equation to get my answer of:

$$d=\frac{1}{2}L(\frac{T}{mg} +1)$$

Does that seem ok?

 

[tiny-tim]

Hi Raziel2701!

(try using the X² icon just above the Reply box )

A pendulum of length L is initially held horizontal, and is then released. The string runs into a peg a distance d below the pivot. What is the smallest value of d for which the string remains taught at all times?

Your answer should not have T in it …

the smallest value of d for which the string remains taut is that for which T = 0 at the top of the loop.

And I think you have the wrong set-up …

the peg is d directly below the pivot, so the top of the loop will be 2d - L below the pivot.