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Chapter 7: Problem 84

When sand is poured on a rotating disc, its angular velocity will (a) decrease (b) increase (c) remain constant (d) None of these

Short Answer

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(a) decrease

Step by step solution

01

Understanding Initial Concepts

First, it's important to understand that angular velocity (\(\omega\)) measures how fast something is rotating. When dealing with rotating objects, the conservation of angular momentum is crucial. Angular momentum (\(L\)) is given by \(L = I \omega\), where \(I\) is the moment of inertia.
02

Applying Conservation of Angular Momentum

When sand is added to a rotating disc, the total mass of the system increases, which affects the moment of inertia. According to the conservation of angular momentum: \(L_{initial} = L_{final}\). Initially, \(I_{initial} \omega_{initial} = I_{final} \omega_{final}\). When sand is added, \(I_{final} > I_{initial}\).
03

Solving for Angular Velocity

Since the angular momentum is conserved and the moment of inertia increases with the added sand, the angular velocity must adjust to maintain the equation. Thus, \(\omega_{final} < \omega_{initial}\), meaning that the angular velocity must decrease to balance the increased moment of inertia.

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Key Concepts

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

Understanding Angular Velocity
Angular velocity, symbolized by \(\omega\), is the rate at which an object rotates around a point or axis. Think of it as the rotational equivalent of linear velocity. Instead of measuring how fast something is moving in a straight line, angular velocity measures how fast something spins around a center.
If you've ever watched a spinning top, its angular velocity is high when it spins fast and low as it slows down. Angular velocity is usually measured in radians per second.Key facts about angular velocity:
  • It tells us the speed of rotation, not the path distance.
  • Angular velocity is vector, meaning it has both magnitude and direction. It points along the axis of rotation.
  • Changes in angular velocity can be influenced by torque or external factors, like adding mass.
Understanding angular velocity is crucial when dealing with rotating systems, as it helps explain how changes in the system affect rotation speed and direction.
How Moment of Inertia Works
Moment of inertia \(I\) is like the rotational equivalent of mass. It tells us how "awkward" or "difficult" it is to change the rotation of an object. More technically, it quantifies the distribution of an object's mass relative to the axis of rotation.
Imagine trying to spin a tightrope walker鈥檚 balancing pole. Its moment of inertia is large, meaning it'll be hard to start or stop rotating.Essential points about moment of inertia:
  • Higher moment of inertia means the object resists changes in angular velocity more.
  • It's dependent not only on mass but also on how that mass is distributed in relation to the rotation axis.
  • The further away the mass is from the axis, the higher the moment of inertia.
The moment of inertia plays a crucial role in the conservation of angular momentum, impacting how changes in a system, like adding sand to a disc, affect rotational speed.
Dynamics of a Rotating Disc
A rotating disc is a common physical system used to demonstrate rotational motion principles, like those involving angular velocity and momentum. When it spins, it exemplifies the concepts we've discussed, such as angular velocity and moment of inertia.
Think of a merry-go-round. When it's spinning, it has angular momentum based on its mass and speed of rotation. Important aspects of a rotating disc:
  • It exemplifies the conservation of angular momentum: Total angular momentum before a change is equal to after, unless an external torque acts on it.
  • Adding mass (like dropping sand) impacts moment of inertia, reducing angular velocity if no external torque is applied.
  • Systems like these help visualize how physics concepts apply in tangible settings.
When we consider pouring sand on a rotating disc, we observe how the increase in moment of inertia results in a decrease in angular velocity to conserve angular momentum.

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Most popular questions from this chapter

A constant power is supplied to a rotating disc. Angular velocity \((\omega)\) of disc varies with number of rotations \((n)\) made by the disc as (a) \(\omega \propto(n)^{1 / 3}\) (b) \(\omega \propto(n)^{3 / 2}\) (c) \(\omega \propto(n)^{2 / 3}\) (d) \(\omega \propto(n)^{2}\)

A body of mass \(M\) and radius \(R\) is rolling horizontally without slipping with speed \(v\). It then rolls up a hill to a maximum height \(h\). If \(h=5 v^{2} / 6 g\), what is the MI of the body? (a) \(\frac{1}{2} M R^{2}\) (b) \(\frac{2}{3} M R^{2}\) (c) \(\frac{3}{4} M R^{2}\) (d) \(\frac{2}{5} M R^{2}\)

Two identical solid cylinders run a race starting from rest at the top of an inclined plane. If one cylinder slides and the other rolls, then (a) the sliding cylinder will reach the bottom first with greater speed (b) the rolling cylinder will reach the bottom first with greater speed (c) both will reach the bottom simultaneously with the same speed (d) both will reach the bottom simultaneously but with different speeds

A solid sphere rolls down two different inclined planes of the same height but of different inclinations (a) in both cases, the speeds and time of descent will be same. (b) the speeds will be same but time of descent will be different. (c) the speeds will be different but time of descent will be same. (d) speeds and time of descent both will be different.

A particle of mass \(m=5\) units is moving with a uniform speed \(v=3 \sqrt{2} m\) in the \(X O Y\) plane along the line \(Y=X\) \(+4\). The magnitude of the angular momentum of the particle about the origin is (a) zero (b) 60 unit (c) \(7.5\) unit (d) \(40 \sqrt{2}\) unit
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