Simple pendulum- ang v at time from mass, length, and ang v at 0 rad

[natasha13100]

Homework Statement

A pendulum consists of a 300 gram mass attached to the end of a string of length 75 cm.
• a) Using the small angle approximation(sinθ ≈ θ), derive an equation for the angular acceleration of the pendulum, assuming that there is no friction.
• b)] The pendulum is given a push,and when it passes through the vertical it is traveling with an angular velocity of 1.3 rad/s in the counterclockwise direction. What is the angular velocity of the pendulum 0.5 second later? At what angle will the pendulum be located?
• c) Re-do part b) using the complex number technique.
• d) The next time the pendulum passes through the vertical, a 600 gram mass is placed directly in its path. Assuming that the ensuing collision be- tween the pendulum and mass is perfetly elastic, what will the amplitude and period of the pendulum be after the collision?
I already solved part a. I am having trouble with part b. Where do I start?

Homework Equations

torque, angular acceleration, moment of inertia, angular velocity, angular displacement, etc.

The Attempt at a Solution

For part a, (I know this part is correct because we did a similar problem in class.
torque=±force x moment arm where + is counterclockwise
net torque=torque due to gravity+torque due to string tension=-mglsinθ+0
m=mass, g=acceleration due to gravity
torque=Iα (moment of inertia times angular acceleration) as well
I=ml²
torque=ml²α
set two equations equal to one another:
-mglsinθ=ml²α
α=-g*sin(θ)/l but I have to use small angle approximation so =-gθ/l

for part b, I really need help with what to do first. (I already drew a picture and FBD.)
I have attempted integrating α with respect to time (t) and using the initial angular velocity (ω_i) as the constant. However, I ran into the problem of multiple variables (ω and θ). I also thought about taking the derivative of the formula for the angular displacement (θ=Acos(Ωt+β)) but I ran into the same problem with amplitude (A) and the constant β. (Ω is angular frequency). Any help is greatly appreciated.

 

[haruspex]

I also thought about taking the derivative of the formula for the angular displacement (θ=Acos(Ωt+β)) but I ran into the same problem with amplitude (A) and the constant β. (Ω is angular frequency).

That should succeed. Please post your working.