Moment of I, Angular momentum, K

[joemama69]

Homework Statement

A ruler of mass m, length l, and width h has a hole drilled into it a short distance x from one end and equidistant from both sides, as shown in the diagram. It is anchored on a frictionless air table by a nail driven through the hole and is then set rotating about the nail. For the following questions, the answers to any part can include the symbols representing the answers to previous parts, whether or not the previous parts were correctly answered. There is no need to substitute the expressions from the previous parts.

a) Find the moment of inertia I of the ruler about the axis of rotation.

b) If the rotation has a period P, what is the magnitude L of the angular momentum of the rotating ruler?

c) Continuing to take P as the period of rotation, what is the kinetic energy K of the ruler?

The ruler is now suspended by a fixed horizontal rod, which passes through the hole drilled in the ruler and forms a frictionless pivot. The ruler is held so that it makes an angle θ to the vertical, as shown in the
diagram, and is then released. Take the acceleration of gravity as g (g > 0).

d) Just after it is released, what is the torque acting on
it? Since the rotation is about a fixed axis, we can use the scalar
definition of torque, with the positive direction taken as counter-
clockwise. (If you prefer to describe the torque as a vector, that
would also be completely acceptable.)

Homework Equations

The Attempt at a Solution

part a) i used the parralele axis theorem

I = I_cm + Mh² = 1/12 M (l² + h²) + M(.5l - x)²

I realize i can simplify but is this correct so far

part b)

is it just L = Iw where it is the I from part a and then w = (2pi)/P

part c)

K = .5Iw² where I is from part a and w is from part b so...K = 2Ipi²/P²

part d) still working on

 

[ideasrule]

part a) i used the parralele axis theorem

I = I_cm + Mh² = 1/12 M (l² + h²) + M(.5l - x)²

I realize i can simplify but is this correct so far

Yes

is it just L = Iw where it is the I from part a and then w = (2pi)/P

Yes

K = .5Iw² where I is from part a and w is from part b so...K = 2Ipi²/P²

Yes.

Have more confidence in yourself!

 

[joemama69]

thanks,

part d)

is it just Toruqe = -mghsinQ

do u do small approx an dmake it simQ=Q