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## Homework Statement

A stick of length 2L and negligible mass has a point mass m affixed to each end. The stick is arranged so that it pivots in a horizontal plane about a frictionless vertical axis through its center. A spring of force constant k is connected to one of the masses. The system is in equilibrium when the spring and stick are perpendicular. The stick is displaced through a small angle theta.

a) Determine the restoring torque when the stick is displaced from equilibrium through the small angle theta

b) Determine the magnitude of the angular acceleration of the stick just after it has been released

c) Write the differential equation whose solution gives the behavior of the system after it has been released

d) Write the expression for the angular displacement theta of the stick as a function of time t after it has been released from rest

theta = displacement angle, phi=angle between stick and spring (complement)

Torque = f X d

f=kLsin(theta)

a) torque = [kLsintheta]*Lsin(phi) = kL2sin(theta)cos(theta)

b) alpha = torque/moment of inertia = [kL2sin(theta)cos(theta)]/[2mL2]

alpha=ksin(theta)cos(theta)/[2m] = k/[4m]*sin(2theta)

Im pretty sure about this, but now what do they mean by "behavior of the system" (displacement?, velocity?) and how do I get things in terms of t?

My attempt:

c) alpha=k/[4m]*sin(2theta) = d2theta/dt2

d2theta/sin(2theta) = k/[4m]*dt2

Am I supposed to integrate that twice? It seems way too intense for an AP to integrate ln[abs(tantheta)]dtheta. Is there a mistake anywhere?

## Homework Equations

Torque = f X d

alpha = torque/moment of inertia

## The Attempt at a Solution

theta = displacement angle, phi=angle between stick and spring (complement)

f=kLsin(theta)

a) torque = [kLsintheta]*Lsin(phi) = kL2sin(theta)cos(theta)

b) alpha = torque/moment of inertia = [kL2sin(theta)cos(theta)]/[2mL2]

alpha=ksin(theta)cos(theta)/[2m] = k/[4m]*sin(2theta)

Im pretty sure about this, but now what do they mean by "behavior of the system" (displacement?, velocity?) and how do I get things in terms of t?

My attempt:

c) alpha=k/[4m]*sin(2theta) = d2theta/dt2

d2theta/sin(2theta) = k/[4m]*dt2

Am I supposed to integrate that twice? It seems way too intense for an AP to integrate ln[abs(tantheta)]dtheta. Is there a mistake anywhere?