Chapter 11 **#33**.

Key equation:

In the question, they say that the clock loses half a minute per day.

Ok, keep this information in mind but lets now consider something else. A grandfather clock goes tick tock tick tock tick tock.. A grandfather clock is bascially a circular disk hang by a string. Thus, it can be considered a pendulum. Lets assign the equilibrium position of the grandfather clock to be a vertical string with a disk on the bottom.

Now, that string with the disk is going to oscillates left and right of that equilibrium position. Suppose we say that when the disk goes to the left of the equilibrium position, it ticks. When the disk goes to the right of the equilibrium position, it tocks. Ok?

Now, I am going to grab on to that string with the disk and put it in the Tick position. You know that every time the clock tick, 1 second has passed and every time that clock tock, another second has passed right?

So, now I am going to release that disk and let it swing. It will swing to the right and go Tock. That's 1 second. Now, it will swing back to the left and go Tick. That's another second, so two seconds in total.

But look, I released at the tick position and it took 2 seconds to swing back the tick position. Thus, it has completed one cycle. Therefore, its period is 2 seconds.

Now we come back to the lose half a minute per day thingie. A day is defined as 24 hr x 60 min x 60 sec. Since each period is 2 s. A day has 24 x 60 x 60 /2 number of periods. So, in 24 x 60 x 60 /2 periods you lose or slowed down half a minute or 30 seconds. This means that every period, you are slowed down by

30 / (24 x 60 x 60 /2) seconds.

This means that you must go faster to counter the loss. Normally, every period is 2 seconds. Since we are losing time here, we must make our period less than 2 seconds to counterbalance the loss. How much less than 2 seconds? Well, we must make our period = 2 - 30 / (24 x 60 x 60 /2). Why? Because we want to keep our period at 2 second. However, since we are slowed down by 30 / (24 x 60 x 60 /2) seconds every period, we must go 30 / (24 x 60 x 60 /2) seconds faster every period. Thus, our "fixed" period is going to be 2 - 30 / (24 x 60 x 60 /2).

Ok, so now we know our T, which is 2 - 30 / (24 x 60 x 60 /2). We have the equation listed above. Plug T in and solve for L. Your answer should be .. 0.9923 m. This means that you must short the string by 0.9930 m - 0.9923 m.

**#47**.

Key equation:

I = P/A = P/(4*pi*r^2) (Intensity = Power/Area) Area here is the surface area of a sphere

Given: I = 2.0 x 10^6 J/m^2

r = 48000 m

Since this wave comes from the same source, the power is going to stay constant no matter how far you run.

Thus, we begin by solving for P . P = I(4*pi*r^2) = (2.0 x 10^6 J/m^2)(4*pi*(48000 m)^2)

Question a) give us a new radius, r = 1000 m and ask for I. We know P and r so I isn't hard to find.

I = P/(4*pi*r^2) = (2.0 x 10^6 J/m^2)(4*pi*(48000 m)^2) / (4*pi*(1000 m)^2)) = 4.6 x 10^9 W/m^2

Question b) Ask us for power (power is defined as Energy / time , hence the rate of that energy passed) given A = 5.0 m^2 and intensity at 1.0 km.

Well, we are still with the same source so intensity at 1.0 km is same as the one we calculated above.

I = 4.6 x 10^9 W/m^2.

A = 5.0 m^2

P = IA = (4.6 x 10^9 W/m^2.)(5.0 m^2) = 2.3 x 10^10 W

**#49 **

Ok, first thing. We know that

They tell us that the two waves have frequencies that are the same. So The ratio of m/k are the same for each wave. (Everything else is constant) However, since these two waves travel on the same sting, we can say that they have the same m and k.

Now, P is defiend as E / t. Energy on a string is 1/2kA^2 .

So, P = (1/2kA^2)/ t

Well, power is really Energy per second so t is just 1.

So really, P = 1/2kA^2

We are told that one wave has three times the power of the other.

Ok so, P1 = 3P1

1/2kA1^2 = 3 * (1/2kA2^2 )

1/2 from both sides cancel out.

kA1^2 = 3 kA2^2

We just said earlier that k is the same for both since they travel on the same string. So k cancel out as well.

A1^2 = 3 A2^2

This means

A1^2/A 2^2 = 3

(A1/A2)^2 = 3

Square root both sides..

A1/A2 = sqrt(3)

That's your answer.