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Chapter 7: Problem 67

A thin rod of mass \(m\) and length \(2 l\) is made to rotate about an axis passing through its centre and perpendicular to it. If its angular velocity changes from 0 to \(\omega\) in time \(t\), the torque acting on it is (a) \(\frac{m l^{2} \omega}{12 t}\) (b) \(\frac{m l^{2} \omega}{3 t}\) (c) \(\frac{m l^{2} \omega}{t}\) (d) \(\frac{4 m l^{2} \omega}{3 t}\)

Short Answer

Expert verified
The correct answer is (b) \( \frac{m l^2 \omega}{3 t} \).

Step by step solution

01

Identify Moment of Inertia

For a rod that rotates about an axis through its center, the moment of inertia is calculated as \( I = \frac{1}{12} m (2l)^2 \). Simplifying this, we get \( I = \frac{1}{3} m l^2 \).
02

Determine Angular Acceleration

Angular acceleration \( \alpha \) is the rate of change of angular velocity. It can be calculated using \( \alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega - 0}{t} = \frac{\omega}{t} \).
03

Calculate Torque

Torque \( \tau \) acting on the rod is given by \( \tau = I \alpha \). Substituting the values of \( I = \frac{1}{3} m l^2 \) and \( \alpha = \frac{\omega}{t} \), the expression for torque becomes \( \tau = \frac{1}{3} m l^2 \cdot \frac{\omega}{t} = \frac{m l^2 \omega}{3 t} \).
04

Match with Choices

Comparing the calculated torque \( \frac{m l^2 \omega}{3 t} \) with the given options, it corresponds to option (b).

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Key Concepts

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

Moment of Inertia
When dealing with rotational motion, the moment of inertia is a key concept that describes how the mass of an object is distributed in relation to the axis it rotates around. Essentially, it's the rotational equivalent of mass in linear motion. For a thin rod rotating about its center, the formula is derived as \( I = \frac{1}{12} m (2l)^2 \), which simplifies to \( I = \frac{1}{3} m l^2 \). This value indicates how "resistant" the rod is to changes in its angular motion. The further the mass is from the axis, the higher the moment of inertia, making it harder to start rotating or to stop.
Angular Acceleration
Angular acceleration, denoted by \( \alpha \), measures how the angular velocity of an object changes with time. In this exercise, it is given by the formula \( \alpha = \frac{\Delta \omega}{\Delta t} \), where \( \Delta \omega \) is the change in angular velocity and \( \Delta t \) is the time it takes for this change.
For the rod in question, the change in angular velocity is from zero to \( \omega \), hence \( \alpha = \frac{\omega}{t} \). This establishes how quickly the rod reaches its target angular velocity, influencing the torque needed to achieve this rotational motion.
Rotational Motion
Rotational motion occurs when an object spins around a fixed axis. It differs from linear motion in that it's concerned with angles, rather than straight line paths. Several aspects are involved in rotational motion, including angular velocity, angular acceleration, and torque.
Understanding rotational motion is crucial for analyzing the dynamics of systems ranging from simple mechanical devices to complex astronomical phenomena. In our exercise, examining how a rod rotates about its center provides practical insights into the balance of forces and moments in rotational dynamics.
Kinematics
Kinematics involves the study of motion without taking into account the forces that cause the motion. In the context of rotational motion, kinematics helps us describe the angular displacement, angular velocity, and angular acceleration.
Just as kinematics in linear motion covers distance, speed, and acceleration, rotational kinematics relates to angles, angular speed, and angular acceleration. These concepts are linked by formulas that allow us to predict how an object will move through space over time, making it possible to determine the torque needed when the angular speed changes as observed in this exercise.
Angular Velocity
Angular velocity, often represented by \( \omega \), describes how fast an object is spinning. It is a vector quantity, indicating not just the speed of rotation but the axis around which it occurs. In the problem at hand, the angular velocity increases from 0 to \( \omega \) as the rod rotates.
It's crucial to understand that angular velocity connects with other rotational concepts like torque and moment of inertia: the change in angular velocity over time demands a certain torque to be applied, which depends on the object's moment of inertia.

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Most popular questions from this chapter

Moment of inertia of a uniform annular disc of internal radius \(r\) and external radis \(R\) and mass \(M\) about an axis through its centre and perpendicular to its plane is (a) \(\frac{1}{2} M\left(R^{2}-r^{2}\right)\) (b) \(\frac{1}{2} M\left(R^{2}+r^{2}\right)\) (c) \(\frac{M\left(R^{4}+r^{4}\right)}{2\left(R^{2}+r^{2}\right)}\) (d) \(\frac{1}{2} \frac{M\left(R^{4}+r^{4}\right)}{\left(R^{2}-r^{2}\right)}\)

The moment of inertia of a metre stick of mass \(300 \mathrm{gm}\), about an axis at right angles to the stick and located at \(30 \mathrm{~cm}\) mark, is (a) \(8.3 \times 10^{5} \mathrm{~g}-\mathrm{cm}^{2}\) (b) \(5.8 \mathrm{~g}-\mathrm{cm}^{2}\) (c) \(3.7 \times 10^{5} \mathrm{~g}-\mathrm{cm}^{2}\) (d) None of these

Generally the mass of a fly wheel is concentrated in its rim. Why? (a) To decrease the moment of inertia (b) To increase the moment of inertia (c) To obtain stable equilibrium (d) To obtain a strong wheel

A uniform cube of side \(a\) and mass \(m\) rests on a rough horizontal table. A horizontal force \(F\) is applied normal to one of the faces at a point that is directly above the centre of face, at a height \(\frac{3 a}{4}\) above the base. The minimum value of \(F\) for which the cube begins to tilt about the edge is (Assume that the cube does not slide) (a) \(\frac{m g}{4}\) (b) \(\frac{2 m g}{3}\) (c) \(\frac{3 m g}{4}\) (d) \(m g\)

A thin hollow cylinder is free to rotate about its geometrical axis. It has a mass of \(8 \mathrm{~kg}\) and a radius of 20 \(\mathrm{cm}\). A rope is wrapped around the cylinder. What force must be exerted along the rope to produce an angular acceleration of \(3 \mathrm{rad} / \mathrm{s}^{2}\) ? (a) \(8.4 \mathrm{~N}\) (b) \(5.8 \mathrm{~N}\) (c) \(4.8 \mathrm{~N}\) (d) None of these
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