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Chapter 9: Problem 60

A disc of mass \(2 \mathrm{~kg}\) and radius \(0.2 \mathrm{~m}\) is rotating with angular velocity \(30 \mathrm{rads}^{-1}\). What is angular veloeity, if a mass of \(0.25 \mathrm{~kg}\) is put on periphery of the disc? [UP SEE 2006] (a) \(24 \mathrm{rads}^{-1}\) (b) \(36 \mathrm{rads}^{-1}\) (c) \(15 \mathrm{rads}^{-1}\) (d) \(26 \mathrm{rads}^{-1}\)

Short Answer

Expert verified
The final angular velocity is 24 radians per second, option (a).

Step by step solution

01

Identify the Given Values

The mass of the disc is given as \( m_1 = 2 \mathrm{~kg} \). The radius of the disc is \( r = 0.2 \mathrm{~m} \). The initial angular velocity is \( \omega_1 = 30 \mathrm{rads}^{-1} \). The additional mass added to the periphery is \( m_2 = 0.25 \mathrm{~kg} \).
02

Calculate Initial Moment of Inertia

The moment of inertia of a disc is given by \( I = \frac{1}{2} m r^2 \). Thus, the initial moment of inertia of the disc is \( I_1 = \frac{1}{2} \times 2 \times (0.2)^2 = 0.04 \mathrm{~kg~m}^2 \).
03

Calculate Moment of Inertia with Additional Mass

When the mass is added at the periphery, its moment of inertia is considered as \( m_2 r^2 \). Therefore, \( I_2 = 0.25 \times (0.2)^2 = 0.01 \mathrm{~kg~m}^2 \). The total moment of inertia is now \( I_{\text{total}} = I_1 + I_2 = 0.04 + 0.01 = 0.05 \mathrm{~kg~m}^2 \).
04

Apply Conservation of Angular Momentum

The conservation of angular momentum states that the initial angular momentum \( L_1 \) should equal the final angular momentum \( L_2 \). Therefore, \( I_1 \omega_1 = I_{\text{total}} \omega_2 \). Substitute the known values: \( 0.04 \times 30 = 0.05 \times \omega_2 \).
05

Solve for Final Angular Velocity

Re-arrange and solve the equation for \( \omega_2 \): \( \omega_2 = \frac{0.04 \times 30}{0.05} = 24 \mathrm{rads}^{-1} \).
06

Choose the Correct Option

Based on the calculation, the final angular velocity \( \omega_2 = 24 \mathrm{rads}^{-1} \). Therefore, the correct option is (a) \(24 \mathrm{rads}^{-1}\).

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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 an essential concept when dealing with rotational motion. It is often called the rotational analog of mass for linear motion. The moment of inertia quantifies how much resistance a body offers against changes in its rotational motion. It's like the rotational version of inertia we experience in linear motion. For simple shapes like discs, the calculation is straightforward. The formula to determine the moment of inertia for a disc is: \[I = \frac{1}{2} m r^2\]where:
  • \( I \) is the moment of inertia,
  • \( m \) is the mass of the disc,
  • \( r \) is the radius of the disc.
When additional mass is added to a rotating body, we have to consider how it affects the moment of inertia. If the mass is added at a distance further from the axis, like the periphery, it increases the moment of inertia significantly.
Angular Velocity
Angular velocity refers to how fast an object rotates or revolves relative to another point, i.e., how quickly the angular position changes over time. It is measured in radians per second (rads/s). Imagine it as the speedometer for rotating objects.Just like velocity in linear motion, angular velocity can change. In our exercise, when the additional mass is placed on the disc, the angular velocity changes. This change is a consequence of the conservation of angular momentum.To calculate the angular velocity, especially when there's a change due to added mass, we use the conservation of angular momentum principle. Initially, the product of the moment of inertia and the angular velocity should equal the final product after any changes to the body:\[I_1 \omega_1 = I_{total} \omega_2\]Solving this equation allows us to understand how the rotational speed adjusts to maintain the conservation laws.
Rotational Motion
Rotational motion is when an object spins around an axis. We're surrounded by examples: from the Earth rotating around its axis to a spinning top. Understanding rotational motion requires considering aspects like moment of inertia and angular velocity. In rotational motion, forces result in a twisting effect known as torque. Torque leads to changes in the state of rotation, just as force leads to changes in linear motion. When other factors, like additional weights, are introduced, the system tends to adjust its properties while conserving angular momentum. This means, if an object becomes heavier or changes shape, it may change its rotational speed but will still preserve its essential dynamics. This conservation in rotational motion is crucial for understanding more complex systems like gyroscopes, engines, and even celestial mechanics.

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Most popular questions from this chapter

Four point masses, each of value \(m\), are placed at the corners of a square \(A B C D\) of side \(l\). The moment of inertia of the system about an axis passing through \(A\) and parallel to \(B D\) is (a) \(\sqrt{3} m l^{2}\) (b) \(3 \mathrm{ml}^{2}\) (c) \(m l^{2}\) (d) \(2 \mathrm{ml}^{2}\)

The moment of inertia of a uniform horizontal cylinder of mass \(M\) about an axis passing through its edge and perpendicular to the axis of the cylinder when its length is 6 times its radius \(R\) is (a) \(\frac{39}{4} M R^{2}\) (b) \(\frac{39}{4} M R\) (c) \(\frac{49}{4} \mathrm{MR}\) (d) \(\frac{49}{4} M R^{2}\)

Of the two eggs those have identical sizes, shapes and weights, one is raw, and other is half boiled. The ratio between the moment of inertia of the raw to the half boiled egg about central axis is (a) one (b) greater than one (c) less than one (d) not comparable

A circular disc of radius \(R\) rolls without slipping along the horizontal surface with constant velocity \(v_{0}\). We consider a point \(A\) on the surface of the disc. Then the acceleration of the point \(A\) is IUP SEE 2005] (a) constant in magnitude as well as in direction (b) constant in direction (c) constant in magnitude (d) constant

Two wheels \(A\) and \(B\) are mounted on the same axle. Moment of inertia of \(A\) is \(6 \mathrm{kgm}^{2}\) and it is rotating at \(600 \mathrm{rpm}\) when \(B\) is at rest. What is moment of inertia of \(B\), if their combined speed is 400 rpm? (a) \(8 \mathrm{~kg} \mathrm{~m}^{2}\) (b) \(4 \mathrm{~kg} \mathrm{~m}^{2}\) (c) \(3 \mathrm{~kg} \mathrm{~m}^{2}\) (d) \(5 \mathrm{~kg} \mathrm{~m}^{2}\)
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