Bullet Hitting Block (Pendulum)

[justadream]

Okay I understand that classic question about whether the height the block reaches will change if the pendulum's length changes (the height stays the same due to energy conservation).

My hypothetical question is:

What about the speed at which the block moves (both normal speed and angular speed)?

As for period:

T = 2pi * sqrt (l / g)

Since l increases, T (the period) increases.

I'm not sure if I can make conclusions about speed (and angular speed) based off of this information.

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[type12]

angular speed is not on the MCAT, but if you must know, the velocity will be the same, but as a consequence, the angular speed will be less.

Analogy: Earth rotates. A person who is closer to the poles completes a circle at the same time as someone on the equator. How is this possible? Answer: their angular frequency is the same because their velocities and acceleration are different.

 

[The Brown Knight]

linear speed depends on velocity of bullet and masses of bullet and block, so it doesn't vary with length of pendulum.
but lengthening the pendulum will DECREASE angular speed:
Intuitively: a longer rope will need to cover fewer degrees to cover the same distance
Mathematically: solving differential equation of pendulum (don't ask me to put that on here) leads to the following expression of angular velocity (which by the way is the same as angular frequency, so you could derive the following by using the relationship T = 2pi/w):
w = sqrt (g/l)
increase l, decrease w

 

[Gauss44]

angular speed is not on the MCAT, but if you must know, the velocity will be the same, but as a consequence, the angular speed will be less.

Analogy: Earth rotates. A person who is closer to the poles completes a circle at the same time as someone on the equator. How is this possible? Answer: their angular frequency is the same because their velocities and acceleration are different.

Why don't you think angular speed is on the MCAT? I have reason to doubt you, but it cannot be mentioned without making a spoiler...

Spoilers below in white text. - DO NOT READ if you have NOT taken all AAMC practice material. Highlight below if you would like to read anyway.

Why do you think angular speed isn't on the MCAT? I just saw a question and passage involving angular frequency on AAMC practice material. I'm not sure if the passage has enough information to answer the question without additional knowledge about w. Feel free to PM me if you want specifics. I am being vague in an attempt NOT to violate SDN policy. (Mods feel free to delete this reply if it is too much information. I tried to supply the least amount of information possible to make my point.)

 

[Lighthill-FFT]

Assuming ideal world (frictionless whatever, weightless string, etc), we still have to ask the question whether or not the bullet is lodged in the pendulum. We have two cases and two ways.

Simple Way that has to be understood before looking at the easy way:

Case 1: Bullet gets lodged (because the combined mass of the system is greater than just the pendulum alone). We first calculate the force of the bullet. Assuming we shot the bullet straight into the pendulum, we can use the simplified formula L=Iw, where I is the moment of inertia, w is the angular frequency and L is the torque. Because the torque is just the force of the bullet, L=f=ma (of bullet). The moment of inertia can be calculated by I=mr^2, where r is the distance from the pivot point. This assumes that the pendulum is effectively a mass point (sometimes it can be treated as such depending on the shape). If you wanted a real world calculation, you need some calculus.

Case 2: The bullet could bounce off (metal pendulum?) completely elastically. Same calculation, we just have to find a way of measuring the momenta of the reflected bullet (not too difficult. Set up a second pendulum and measure how high it goes to calculate the KE, which gives us velocity and thus momentum).

Easy Way:

Case 1: We calculated the linear momenta and translate it into angular momenta, which we use to calculate angular velocity. This approach requires an extremely detailed explanation of physics and a lot of really delicate arguments to fully understand, so the simple way is the preferred way unless you really want to get into the knitty gritty.

Case 2: Same thing, same way.

Physicists Way:

We calculate the Lagrangian or the Hamiltonian of the system and use that to calculate angular momenta. This relies on even more delicate arguments and shouldn't be used at all unless you took a class in dynamics, but is the most powerful way of doing things because its magical (really no other word to describe it). In this specific case, it would be easier to either do it the Easy way or the Simple way.