Rotational Motion of a Hanging Mass

[interxavier]

Homework Statement

Consider a ball of mass m on the end of a string of length l. It hangs from a frictionless pivot. The ball is pulled out so that the string makes an angle thetai with the vertical and is then released.
a. Find w (angular velocity) as a function of the angle the strings makes with the vertical. (Hint: Use conservation of energy.)
b. Find the angular momentum of the ball using |L| = ml^2w
c. Show that t = dL/dt by differentiating L and finding t from its definition.

Homework Equations

U1 + K1 = U2 + K2
V = rw

The Attempt at a Solution

for a:

The answer, according to the book, is w = sqrt(2g/l*(cos(theta) - cos(thetai)))

I used the conservation of energy and got w = sqrt(2g/l*(1 - cos(thetai))). I'm lost.. I don't know where the book got cos(theta)

 

[rock.freak667]

at an angle θ_i, find the energy at that point.

Now this energy is converted into gravitational pe and ke at an angle θ. Find this energy here.


Equate the two.

 

[kerimek]

from.



Hello, thank you for sharing your work. I can see that you are on the right track in using the conservation of energy to solve for the angular velocity of the ball. However, there seems to be a small error in your calculation.

When using the conservation of energy, we need to consider the potential energy at both the initial and final positions. In this case, the initial potential energy is mgh (where h is the height of the ball) and the final potential energy is mgl(1-cosθ). Therefore, the equation should be:

mgh = mgl(1-cosθ) + 1/2Iw^2

where I is the moment of inertia of the ball (which can be approximated as ml^2 for a small, solid ball).

Solving for w from this equation, we get:

w = sqrt(2g/l*(cosθ - cosθi))

This is the same answer as the one given in the book. The difference is the initial potential energy term.

I hope this helps clarify the solution for part a. For parts b and c, you can use the formula for angular momentum (L = Iw) and differentiate it with respect to time to show that it is equal to the torque (t) acting on the ball.

I hope this helps. Keep up the good work!