Question

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

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

Step-by-step solution

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

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)

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.

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)

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.

Key concepts

Angular Speed

When thinking about angular speed, imagine a spinning merry-go-round. It's not just about how fast it spins, but also how this speed is expressed in terms of angles. Angular speed refers to how fast an object rotates or revolves relative to another point, often an axis. In the context of this problem, it's the rate at which the platform rotates.

In mathematical terms, angular speed, often denoted by the Greek letter omega (\( \omega \)), is calculated by converting rotational speed into radians per second. One complete revolution equals \(2\pi \) radians. Hence, to find the angular speed from revolutions per second (rev/s), use this formula:

• \(\omega = \text{Rotational speed} \times 2\pi \)
• For example, \(\omega = 0.020 \text{ rev/s} \times 2\pi \)

Hence, you have \(\omega \approx 0.126\) rad/s. This angular speed determines the behavior of objects attached and rotating with a platform, such as the pendulum in our problem.

Pendulum Motion

Pendulum motion is a classic example of oscillatory motion which, in a static setting, would simply swing back and forth in a straight line. However, when placed on a moving platform, the pendulum's behavior changes. Here, the forces acting on it include both gravity and centripetal force.

The centripetal force pulls the pendulum towards the center of rotation, causing it to hang at an angle when in motion. This creates a new balance between forces:

• The gravitational force acting downward.
• The centripetal force acting horizontally toward the center of the merry-go-round.

These combined forces cause the pendulum to hang at a specific angle, \( \theta \), relative to the vertical. Analyzing this setup requires understanding the component forces and angles, using trigonometric functions like sine and cosine to describe them. Solving these gives you insight into the angle, thereby explaining how motion alters the pendulum's positioning.

Rotation Axis

The rotation axis is an imaginary line around which rotation occurs. It serves as a central point for rotation or revolution, affecting how other objects, such as a pendulum, behave concerning it.

In practical terms, think of the rotation axis as the pole on a merry-go-round. Anything attached or moving with this platform must contend with the centripetal forces directed inward, perpendicular to the direction of gravitational force. The distance from the rotation axis, also called radius (r), greatly influences the strength of this force. For a pendulum held 6 meters from this axis, it contributes to the calculation of centripetal force.

• Centripetal force formula: \(Fc = m \times r \times \omega^2\)
• Where \(r\) is the distance (\(6.0\, \text{m}\) in our example)

Recognizing how objects behave concerning a rotation axis is crucial in solving problems of rotational dynamics like this one, emphasizing how distances and rotational speed interact to define motion.