C Pendulum. A small rock with mass 0.12 kg is fastened to a massless string with length 0.80 m

C Pendulum. A small rock with mass 0.12 kg is fastened to a...

C Pendulum. A small rock with mass $0.12 \mathrm{~kg}$ is fastened to a massless string with length $0.80 \mathrm{~m}$ to form a pendulum. The pendulum is swinging so as to make a maximum angle of $45^{\circ}$ with the vertical. Air resistance is negligible. (a) What is the speed of the rock when the string passes through the vertical position? What is the tension in the string (b) when it makes an angle of $45^{\circ}$ with the vertical, (c) as it passes through the vertical?

Transcript

00:01 So we have a pendulum that is brought back by an angle of 45 degrees to the vertical.

00:06 And the pendulum has a length called this l of 0 .8 meters.

00:11 And that mass is fastened to it at the end of 120 grams.

00:18 And part a asks us, what is the speed of the rock when the string passes through the vertical position? so let's say this is right here.

00:26 What is the speed at this point? so let's look at first off the initial potential energy of this system because this is a conservation of energy problem.

00:37 So the initial potential energy, let's define zero potential energy as being at the tether location of the string to the ceiling.

00:45 All our potential energies are measured below that.

00:48 They're going to be negative.

00:49 So it'll be negative mg times l times we're looking at the vertical component of this.

00:57 So l times the cosine of theta, which theta is 45.

01:02 We'll just say that.

01:02 And then the final potential energy is just going to be negative mgl.

01:07 So we can see the change in potential energy, the final minus the initial, is going to be like mgl, should be cosine of theta minus 1.

01:21 If i'm looking at that correctly.

01:24 So that's our change in potential energy.

01:26 So what we have then is the change in kinetic energy is equal to the negative change.

01:29 Change in potential energy.

01:31 So we'll have in one half mb square, which is our final kinetic energy, is equal to m gl times one minus the cosine of theta.

01:40 The reason the minus signs got switched around is because we have a negative sign right here.

01:45 So the mass is canceled.

01:46 We can see our velocity then becomes 2gl times one minus the cosine of theta.

01:54 And so when we plug in our values, you know, this is the square root of two times nine point eight...

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