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Chapter 10: Problem 54

A boy is standing on a platform which is free to rotate about its axis. The boy holds an open umbrella in his hand. The axis of the umbrella coincides with that of the platform. The moment of inertia of "the platform plus the boy system" is \(3 \cdot 0 \times 10^{-3} \mathrm{~kg}-\mathrm{m}^{2}\) and that of the umbrella is \(2 \cdot 0 \times 10^{-3} \mathrm{~kg}-\mathrm{m}^{2}\). The boy starts spinning the umbrella about the axis at an angular speed of \(2 \cdot 0\) rev/s with respect to himself. Find the angular velocity imparted to the platform.

Short Answer

Expert verified
The angular velocity imparted to the platform is \(\frac{8\pi}{3}\) rad/s.

Step by step solution

01

Understand the Concept of Angular Momentum Conservation

The system described is isolated, meaning no external torques act on it. Therefore, the total angular momentum must be conserved. The initial angular momentum is zero, as neither the platform nor the umbrella is rotating relative to the ground initially.
02

Write the Angular Momentum Conservation Equation

When the boy spins the umbrella at an angular speed of \( \omega_{u} = 2 \cdot 0 \) rev/s (convert to rad/s by multiplying with \(2\pi\)), the platform will start rotating in the opposite direction to conserve angular momentum.Let \( \omega_{p} \) be the angular velocity of the platform.Applying angular momentum conservation: \[ (I_{b+p} + I_{u}) \cdot 0 = I_{b+p} \cdot \omega_p - I_{u} \cdot \omega_{u} \]where\(I_{b+p} = 3 \cdot 0 \times 10^{-3} \text{ kg} \cdot \text{m}^2\) for the boy plus platform,\(I_{u} = 2 \cdot 0 \times 10^{-3} \text{ kg} \cdot \text{m}^2 \) for the umbrella.
03

Simplify and Solve for the Platform Angular Velocity

Since the initial total angular momentum is zero:\[ I_{b+p} \cdot \omega_p = I_{u} \cdot \omega_{u} \]Substitute the known values and solve:\[3 \cdot 0 \times 10^{-3} \ \omega_p = 2 \cdot 0 \times 10^{-3} \cdot 2 \cdot 0 \times 2\pi \]Simplifying, we find:\[ \omega_p = \frac{2 \cdot 0 \times 10^{-3} \times 2 \cdot 0 \times 2\pi}{3 \cdot 0 \times 10^{-3}} = \frac{8\pi}{3} \text{ rad/s} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Moment of Inertia
The moment of inertia is crucial in understanding rotational motion. It measures an object's resistance to changing its rotational speed. Think of it as the rotational equivalent to mass in linear motion.
For a rotating object, moment of inertia depends on the mass distribution relative to the rotation axis.
Different shapes and mass distributions yield different moments of inertia, influencing how easily they rotate.
  • The greater the moment of inertia, the harder it is to change the object's rotation.
  • In our exercise, the moment of inertia for the boy and platform is different from the umbrella because they have distinct mass distributions.
Understanding this concept helps explain why even small changes in rotation involve energy and why objects like gyroscopes can maintain their orientation.
Rotational Motion
Rotational motion refers to the movement of an object around a specific axis. Unlike straight-line (linear) motion, rotational motion involves twisting or spinning action.
In this exercise, the platform and the umbrella involve rotational dynamics, as they rotate about a central point.
  • Rotational motion relies heavily on angular momentum, which is the product of the moment of inertia and angular velocity.
  • Angular momentum conservation is a key principle; in a closed system, the total angular momentum remains unchanged unless acted on by external torques.
For our boy and platform scenario, the angular momentum before and after the boy spins the umbrella must be the same. This law allows us to predict how other parts of the system (like the platform) will react.
Adjusting one component's motion alters another to maintain the rotational balance.
Angular Velocity
Angular velocity is like the speed of rotation. It's a vector that describes how quickly an object rotates around its axis. Measured in radians per second, angular velocity tells us the rate of rotation.
  • When the boy spins the umbrella, he imparts angular velocity to it, causing rotation about its axis.
  • This action affects the angular velocity of the platform, as shown by the conservation of angular momentum.
  • When the umbrella spins faster or slower, it can change the rotation rate of the platform accordingly.
In essence, angular velocity gives us insight into motion dynamics and helps predict outcomes in rotating systems like gears or flywheels.
It's essential to convert between units, like from revolutions per second to radians per second, for accurate calculations in physics problems.

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Most popular questions from this chapter

A uniform metre stick of mass \(200 \mathrm{~g}\) is suspended from the ceiling through two vertical strings of equal lengths fixed at the ends. A small object of mass \(20 \mathrm{~g}\) is placed on the stick at a distance of \(70 \mathrm{~cm}\) from the left end. Find the tensions in the two strings.

Because of the friction between the water in oceans with the earth's surface, the rotational kinetic energy of the earth is continuously decreasing. If the earth's angular speed decreases by \(0-0016 \mathrm{rad} /\) day in 100 years, find the average torque of the friction on the earth. Radius of the earth is \(6400 \mathrm{~km}\) and its mass is \(6 \cdot 0 \times 10^{24} \mathrm{~kg}\).

A thin spherical shell lying on a rough horizontal surface is hit by a cue in such a way that the line of action passes through the centre of the shell. As a result, the shell starts moving with a linear speed \(v\) without any initial angular velocity. Find the linear speed of the shell after it starts pure rolling on the surface.

A cylinder rolls on a horizontal plane surface. If the speed of the centre is \(25 \mathrm{~m} / \mathrm{s}\), what is the speed of the highest point?

A rod of mass \(m\) and length \(L\), lying horizontally, is free to rotate about a vertical axis through its centre. \(\mathrm{A}\) horizontal force of constant magnitude \(F\) acts on the rod at a distance of \(L / 4\) from the centre. The force is always perpendicular to the rod. Find the angle rotated by the rod during the time \(t\) after the motion starts.
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