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Chapter 9: Problem 74

A girl on a merry-go-round platform holds a pendulum in her hand. The pendulum is \(6.0 \mathrm{~m}\) from the rotation axis of the platform. The rotational speed of the platform is 0.020 rev/s. It is found that the pendulum hangs at an angle \(\theta\) to the vertical. Find \(\theta\)

Short Answer

Expert verified
Answer: The pendulum hangs at approximately 0.279° to the vertical in the merry-go-round platform.

Step by step solution

01

Convert the Rotational Speed to Angular Speed

First, we need to convert the given rotational speed (0.020 rev/s) to angular speed in radian per second. Angular speed (ω) = Rotational speed × 2π ω = 0.020 rev/s × 2π rad/rev ω ≈ 0.126 rad/s
02

Find the Centripetal Force

Next, let's find the centripetal force (Fc) acting on the pendulum. The centripetal force formula is: Fc = m × r × ω^2 where m is the mass of the pendulum, r is the distance from the rotation axis, and ω is the angular speed. We don't know the mass of the pendulum. However, we will see that it will cancel out in the final equation. So we can keep 'm' in our calculations. The centripetal force will be: Fc = m × 6.0 m × (0.126 rad/s)^2 Fc ≈ 0.048m N (approximately)
03

Analyze the Forces Acting on the Pendulum

Now, let's analyze the forces acting on the pendulum. There are two forces acting on it: gravity (Fg = m × g) and centripetal force (Fc). These two forces together cause the pendulum to hang at an angle θ to the vertical.
04

Write the Equations for the Forces in the x and y Directions

The centripetal force acts horizontally, while the gravitational force acts vertically. So we can write the equations for the forces in the x and y directions. For the x-direction: Horizontal Force = Fc × sin(θ) 0.048m = m × g × sin(θ) For the y-direction: Vertical Force = Fg × cos(θ) m × g = m × g × cos(θ)
05

Solve for the Angle θ

We can now solve for the angle θ. From the x-direction equation, we get: sin(θ) = 0.048m/mg sin(θ) = 0.048/g From the y-direction equation, we get: cos(θ) = m×g/m×g cos(θ) = 1 Now, divide the sin(θ) equation by the cos(θ) equation to get tan(θ): tan(θ) = sin(θ)/cos(θ) tan(θ) = (0.048/g) / 1 θ = arctan(0.048/g) Assuming g = 9.81 m/s^2, we can find the angle θ: θ = arctan(0.048/9.81) θ ≈ 0.279° So, the pendulum hangs at approximately 0.279° to the vertical in the merry-go-round platform.

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