Rotational Momentum Problem Help

[xnitexlitex]

A ballistic pendulum consists of an arm of mass M and length L = 0.55 m. One end of the arm is pivoted so that the arm rotates freely in a vertical plane. Initially, the arm is motionless and hangs vertically from the pivot point. A projectile of the same mass M hits the lower end of the arm with a horizontal velocity of V = 1.8 m/s. The projectile remains stuck to the free end of the arm during their subsequent motion. Find the maximum angle to which the arm and attached mass will swing in each case.

a) The arm is treated as an ideal pendulum, with all of its mass concentrated as a point mass at the free end.

b) The arm is treated as a thin rigid rod, with its mass evenly distributed along its length.

Relevant equations:
mvL = (I_rod + I_proj)ω_f
E_i = E_f
There might be a simpler equation, though.

Here's how I tried to solve it:
First, I solved for ω_f by substituting the moments of inertia. I ended up with this:

ω_f = mvL / [(1/3)mL² + mL²] , which became:
3mv/(mL+3m).

Then, I used energy conservation, resulting in this equation:
1/2(I_rod + I_proj)ω_f² + mg(L/2) = mg(L-L cos θ) + mg{L/2 + [(L/2) - (L/2 cos θ)]}

I canceled the masses, yet I still got it wrong. I can't seem to find a way to simplify the equation! All the above is for part B. If I can get part B, I can get A, since it's the same, but with one less moment of inertia to worry about. All help is appreciated.

I'm sorry if this post isn't perfect. This is my first post here.

 

[grzz]

1st stage: initially both projectile and pendulum are at rest.
2nd stage: the projectile hits and gets embedded in the pendulum.
3rd stage: both move as one body to a final angle.

I do not think that energy is conserved from stage (1) to stage (3) as shown above because in the impact at stage (2) energy is 'lost'.

 

[xnitexlitex]

1st stage: initially both projectile and pendulum are at rest.
2nd stage: the projectile hits and gets embedded in the pendulum.
3rd stage: both move as one body to a final angle.

I do not think that energy is conserved from stage (1) to stage (3) as shown above because in the impact at stage (2) energy is 'lost'.

Okay, but using momentum conservation doesn't get me the angle. Do I need Newton's Second Law?

 

[grzz]

Momentum conservation will give the speed of the combined system just after impact. THEN conservation of energy can be used and this will give the angle required.

 

[asteriatic]

Dear scientist,

Thank you for your question and for providing your work and thought process so far. It seems like you are on the right track in solving this rotational momentum problem.

One suggestion I have is to check your units and make sure they are consistent throughout your calculations. For example, in your first equation, the units for ωf should be in radians/second, but the units for the moments of inertia are in kg*m^2, so there may be an error in your calculation there.

Additionally, in your energy conservation equation, you can simplify the right side by combining the terms with cos θ. This will result in a quadratic equation, which you can solve for θ using the quadratic formula.

I hope this helps and good luck with your problem-solving! Keep in mind that it's always helpful to double check your units and equations to make sure they are consistent and accurate.