You are exploring a newly discovered planet. The radius of...

You are exploring a newly discovered planet. The radius of the planet is $7.20 \times 10^{7} \mathrm{~m}$. You suspend a lead weight from the lower end of a light string that is $4.00 \mathrm{~m}$ long and has mass $0.0280 \mathrm{~kg}$. You measure that it takes $0.0685 \mathrm{~s}$ for a transverse pulse to travel from the lower end to the upper end of the string. On the earth, for the same string and lead weight, it takes $0.0390 \mathrm{~s}$ for a transverse pulse to travel the length of the string. The weight of the string is small enough that you ignore its effect on the tension in the string. Assuming that the mass of the planet is distributed with spherical symmetry, what is its mass?

Key Concepts

Wave Propagation in Strings

This concept involves understanding how transverse waves travel along a string, where the wave speed is determined by the tension in the string and its linear mass density according to the relation v = ?(T/?). This principle is used to connect measured transit times of pulses along the string to the forces present in the system.

Tension from Gravitational Forces

In a vertical setup, the tension in the string is provided by the weight of the suspended mass, calculated as T = mg. This relation allows one to connect the local gravitational acceleration with the tension in the string, enabling indirect measurements of gravity through observations of wave speed along the string.

Gravitational Field and Planetary Mass

The gravitational acceleration at the surface of a spherical planet is given by g = GM/R², where G is the gravitational constant, M is the planet's mass, and R is its radius. By determining the local value of g (inferred through wave propagation in the string) and knowing the planetary radius, one can calculate the mass of the planet assuming a spherically symmetric mass distribution.

Transcript

00:01 So this is a very interesting question because we're trying to find this mass of the planet.

00:04 We'll denote it as m sub p.

00:07 And we have some information.

00:10 However, we need to use strings and time periods in order to approximate the mass of this planet.

00:20 So let's just go right into it.

00:23 We'll circle that.

00:25 So we have the speed of propagation of a wave, and this is going to be equal to.

00:30 The force, rather the force tension in the string, times the linear density of the string.

00:36 Now, we're going to use something different for f, and we can say that f is going to be equal to m .a.

00:45 Because this is due to newton's second law of motion.

00:49 And mu can be rewritten as the mass of the string divided by the length of, divided by thy length.

00:57 So let's simplify this and we'll have lma over m.

01:06 So this is a total mass and this is simply a mass of a string.

01:11 So let's try to relate this.

01:17 If this is the speed of propagation of a wave, we can say that lma equals m, but we can also say that this is simply equal to a length over a time.

01:31 Which we can do this, we can make this relationship because these are simply saying the exact same thing.

01:41 And at this point, let's solve for acceleration.

01:47 At this point, we can say that acceleration is going to be equal to m large m over l small m.

01:56 And then, of course, l squared over t squared.

02:05 Let's, the only acceleration, the acceleration that we're trying to account for here essentially is gravity.

02:14 So let's set this equal to gravity and simplify this equation here.

02:20 So we can say that ml over mt squared equals g.

02:28 At this point, we can say, okay, let's solve for this mass right here, because that's the mass that we are really looking for.

02:39 So, we can say, okay, m is going to be equal to ml over gt squared, the acceleration due to gravity times the time period squared.

02:50 And then we can further say that the time taken, let's further define these variables and say that m equals m.

02:58 M l over gt squared and then we're going to say e.

03:05 So t sub e is simply the time taken for a wave pulse to propagate on earth...

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