# Posts

## Demo Problems: Then and Now

This upcoming week’s demo problem on relativity is:

An armada of spaceships that is 1.00 ly long (in its rest frame) moves with a speed 0.800c relative to a ground station in frame S. A messenger travels from the rear of the armada to the front with a speed of 0.950c relative to S. How long does the trip take as measured (a) in the messenger’s rest frame, (b) in the armada’s rest frame, and (c) by an observer in frame S?

To see the solution to last week’s demo problem, see the following file: Demo Problem 10-25

A write-up of how to do venturi tubes and heat engines is still in the works.

## How to: Demo problem 10/21

Question: A uniform trap door to a cellar lies horizontal on a floor.  The trap door has a mass of 7 kg and a length of 90 cm measured from the hinge.  There is a solid spherical 5 kg knob of radius 4 cm attached to the trap door 80 cm from the hinge.  Someone tries to open the trap door by lifting at an angle of 15 degrees from vertical with a force of 110 N.  What is the angular acceleration of the trap door?

Instead of spending a bunch of time trying to format the math into html for the post, I’ve written it out in a pdf. Click the link below to see the file explaining how to do the problem!

Demo10-21

## Physics and History: The Ballistic Pendulum

In last week’s recitation question (scroll down two posts if you don’t know what I’m talking about), we examined a bullet (mass m) hitting a block (mass M) hanging from a string (length L). At the end I ask you what would have happened if there was an inelastic collision and the bullet was stuck in the block. Can you find it’s initial velocity now? I encourage you to go back and try this problem if you haven’t already. You should be able to use conservation of energy to get vi=[1+(m/M)]√2gh, where h is how high the pendulum swung.

Find the initial velocity based on the height which the pendulum reached.

You might be asking yourself, why do we care? Why am I solving this problem? Will this ever matter to anyone?

YES. We did not invent this problem out of thin air, this problem comes from an actual device.

The “ballistic pendulum” was invented in 1742 as a device for testing the firepower of military weapons. If you are about to mass produce 500,000 muskets for your growing empire, you will probably want to know some quantitative numbers about the weapons you’re producing. This is a time before high-speed cameras, before radar-guns, and before precise clocks – how else would you go about trying to accurately measure the speed of a bullet?

An actual ballistic pendulum.

This problem is especially important when it comes to knowing the velocity of artillery shells. When we solve a 2D projectile motion problem, we call it a “cannon problem.” The techniques to solve this problem were developed by real people needed to shoot real cannons at real targets – often shooting over the heads of their own comrades. How confident do you feel about solving a cannon problem on an exam? How confident do you think you would feel solving the same question, while under enemy fire, knowing that you are about to destroy an area the size of a football field? You’d probably feel more confident if you knew that the initial velocity was measured accurately, which is exactly why they invented the ballistic pendulum.

## Demo Problem: 10/21/14

This is the demo problem that we’ll be discussing on Tuesday. I encourage you to give it a shot before coming to recitation.

Question: A uniform trap door to a cellar lies horizontal on a floor.  The trap door has a mass of 7 kg and a length of 90 cm measured from the hinge.  There is a solid spherical 5 kg knob of radius 4 cm attached to the trap door 80 cm from the hinge.  Someone tries to open the trap door by lifting at an angle of 15 degrees from vertical with a force of 110 N.  What is the angular acceleration of the trap door?

## Demo 10/14

Write this out and try it for yourself in your notebook!!! You won’t fully absorb how to do it if you simply skim the solution.

Question: A bullet of mass m and speed v passes completely through a pendulum bob of mass M as shown. The bullet emerges with a speed of v/2. The pendulum bob is suspended by a stiff rod of length L and negligible mass. What is the minimum value of v such that the pendulum bob will barely swing through a complete vertical circle?

First off all, let’s draw out what this question is actually describing in the before/after situations.

The question is asking what the minimum velocity required to make the mass go around in a circle once. That means that it should just *barely* get to the top, so we can assume mass M has no velocity at the top. Let’s consider the conservation of momentum for this situation to see how fast the block M moves after the bullet goes through it.

∑Pbefore = ∑Pafter
mv1 = Mv2 + m(v1/2)

Now, solve for v2.
mv1 – (1/2)mv1 = Mv2
(1/2)mv1 = Mv2
(1/2)(m/M)v1 = v2

Great, now we know how fast the block will move based on the speed that the bullet hits it. Now we want to find out how fast the block needs to move in order to have enough energy to get into position. Consider what the block is doing:

If we call the bottom position our y=0 point, then the top is just height h=2L. At the bottom, there’s no potential energy, and later the block just *barely* passes the top (so v at the top  is zero as well).

This means that the conservation of energy gives
(1/2)Mv22 = Mg(2L)

Simplifying a bit…
(1/2)v22 =2gL
v22 =4gL
v2=√4gL

Now you can throw in the result from earlier to sub out v2 in terms of v1, and then solve for v1
(1/2)(m/M)v1= √4gL
(1/2)(m/M)v1= 2√gL
v1= 4(M/m)√gL

And you’re done!!!

Now, consider this: If the bullet hit the pendulum in an inelastic collision, would the velocity needed be greater than, less than, or equal to the previous velocity we found?