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$\bullet$ $\bullet$ On the planet Newtonia, a simple pendulum having a bob with a mass of 1.25 $\mathrm{kg}$ and a length of 185.0 $\mathrm{cm}$ takes 1.42 $\mathrm{s}$ , when released from rest, to swing through an angle of $12.5^{\circ}$ , where it again has zero speed. The circumference of Newtonia is measured to be $51,400 \mathrm{km} .$ What is the mass of the planet Newtonia?

9$.08 \times 10^{24} \mathrm{kg}$

Physics 101 Mechanics

Chapter 11

Elasticity and Periodic Motion

Equilibrium and Elasticity

Periodic Motion

University of Michigan - Ann Arbor

University of Washington

Simon Fraser University

McMaster University

Lectures

04:12

In physics, potential energy is the energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors. The unit for energy in the International System of Units is the joule (J). One joule can be defined as the work required to produce one newton of force, or one newton times one metre. Potential energy is the energy of an object. It is the energy by virtue of an object's position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass is performed by a force which works against the force field of the potential. The potential energy of an object is the energy it possesses due to its position relative to other objects. It is said to be stored in the field. For example, a book lying on a table has a large amount of potential energy (it is said to be at a high potential energy) relative to the ground, which has a much lower potential energy. The book will gain potential energy if it is lifted off the table and held above the ground. The same book has less potential energy when on the ground than it did while on the table. If the book is dropped from a height, it gains kinetic energy, but loses a larger amount of potential energy, as it is now at a lower potential energy than before it was dropped.

02:18

In physics, an oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The oscillation may be periodic or aperiodic.

05:55

On the planet Newtonia, a …

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02:27

CP On the planet Newtonia,…

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The period of a simple pen…

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The period of a pendulum i…

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A pendulum has a period of…

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before we begin, let's write down some useful formulas that we'll need in the future. We know that Newton's law of gravity tells us that the gravitational acceleration is equal to Capital G, which is the gravitational constant times the mass over radius squared. Now we're giving the circumference, and eventually we'll need the radius. So we'll need the relationship between the two, which is that the circumference is two pi r. And then now we're going to start by calculating the period of the motion. Now it gives us half cycle the time to do half cycle, which is one swing. But we want the period, which is the time for a full cycle, which is a swing back and then a swaying back and forth. So the period will be two times the time they give us, which is one point for two, and so the full period is 2.4 seconds. Now we can use the formula that tea is equal to pi Times square root of Ella Virgie Vera. What ji is so solving. Fergie gives cheese eagle two to pie over tea squared times the length of the pendulum, now plugging in T, employing the likes of pendulum gives the gravitational constant of 9.6 meters per second squared for this planet. So now when we go back here and we saw for the mass, we get g r squared over Peggy. And now we need to plug in here. And before you plug in are we need to figure out what it is we're given. The circumference O. R is equal to two pi oversea, which is equal to 8.18 times 10 to the six meters. Yes. Now this is what we're gonna plug in here for. Junior, we're gonna plug in this And in Capital Ji is just a constant equal to 6.673 times 10 to the negative 11. And so the top is 9.6 and ah ra's 8.18 times 10 26 and we have to square that. And so this is what the mass of the planet is in kilograms ends up being 9.0, eh? Times 10 to the 24 children's. And that's the answer

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