Finding mass of Moon with Pendulum Bob

In summary, the conversation discusses the possibility of determining the mass of a planet or large moon using a 'pendulum bob' horizontal circular motion experiment. A cosmonaut on the moon is able to calculate the mass of the moon by measuring the angle and period of a pendulum bob in motion. The conversation also mentions using the pendulum equation and the small angle approximation to derive the gravitational acceleration of the moon. However, there is some confusion about the equations used and the final calculation of the moon's mass.
  • #1
Carterr
3
0

Homework Statement



It is possible to determine the mass of a planet or large moon by using a ‘pendulum bob’ horizontal circular motion experiment. A cosmonaut on the moon finds that a 25 cm long ‘pendulum bob’ moving in uniform horizontal circular motion makes an angle of 22° to the vertical and moves with a period of 2.3 s. What is the mass of the moon? (The radiusof the moon is 1.5 х 105 m)

Homework Equations



??

The Attempt at a Solution



0.25sin(22) = 0.094 (radius)

v = (2∏r) / 2.3 = 0.257 ms-1
 
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  • #2
Start with forces acting on the bob.
 
  • #3
so I tried ac = 0.2572 / 0.094 = 0.7 ms-2

tan(90-22) = g / 0.7
g = 1.12

1.12 = (6.67 x 10-11)M / (1.5 x 105)2

and I got M = 3.78 x 1020 which is wrong.


I'm not sure how to go with this one
 
  • #4
You could use the pendulum equation.

Since 22.3 degree is small enough, use the small angle approx, sin θ ≈ θ so you could derive from the pendulum equation T ≈ 2π√(l/g). So measure the period of the pendulum and find g of the moon. Since g = Gm/r2, calculate m since you know r.
 
  • #5
Where does tan(90 - 22) = g / 0.7 come from?
 
  • #6
you have 22 degrees as the top angle and the Ac in a right angled triangle. 90-22 = 58. Is that right, I'm not sure.

tan(58) = g / 0.7
 
  • #7
First of all, 90 - 22 = 68, not 58. Even then, I do not understand where your equation comes from. What equations do you get from the free body diagram?
 

1. How do you find the mass of the Moon using a pendulum bob?

To find the mass of the Moon using a pendulum bob, you will need to measure the length of the pendulum and the period of its swing. Then, using the formula T = 2π√(L/g), where T is the period, L is the length, and g is the acceleration due to gravity, you can solve for the value of g. Once you have the value of g, you can use Newton's law of universal gravitation, F = (GmM)/r², where F is the force of gravity, G is the gravitational constant, m is the mass of the pendulum bob, M is the mass of the Moon, and r is the distance between the centers of mass of the pendulum bob and the Moon. Rearrange this equation to solve for the mass of the Moon, M.

2. How accurate is this method of finding the mass of the Moon?

This method of finding the mass of the Moon using a pendulum bob is relatively accurate, but it does have some limitations. The accuracy of the measurement depends on the precision of the length and period measurements, as well as the assumptions made in the equations used. Additionally, factors such as air resistance and external forces may affect the accuracy of the measurements.

3. Can this method be used to find the mass of other celestial bodies?

Yes, this method can be used to find the mass of other celestial bodies as long as the necessary measurements can be obtained. The same principles and equations can be applied, but the values of g, G, and r will vary depending on the specific celestial body being measured.

4. Are there alternative methods for finding the mass of the Moon?

Yes, there are alternative methods for finding the mass of the Moon, such as using the motion of natural satellites or measuring the deflection of laser beams. Each method has its own advantages and limitations, and the most appropriate method will depend on the available resources and the desired level of accuracy.

5. Why is it important to know the mass of the Moon?

Knowing the mass of the Moon is important for various reasons. It helps us understand the dynamics of the Earth-Moon system, such as the tides and the Moon's influence on Earth's rotation. It also provides valuable information for space exploration and navigation, as well as for studying the origins and evolution of the Moon and the solar system as a whole.

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