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Chapter 8: Problem 38

When a wheel moves a distance shorter than \(2 \pi R\) while making one rotation, then: (a) \(v_{\mathrm{cm}}R \omega\) (d) \(v_{\mathrm{cm}}>R \omega\)

Short Answer

Expert verified
(a) \(v_{\mathrm{cm}} < R \omega\) or (b) \(v_{\mathrm{cm}} < R \omega\) - equivalent options.

Step by step solution

01

Understanding the problem

We are given a scenario where a wheel moves a distance shorter than its circumference (which is \(2 \pi R\)) while completing one rotation. Our task is to determine the relationship between the center of mass velocity \(v_{\mathrm{cm}}\) and the product of radius \(R\) and angular velocity \(\omega\).
02

Concepts needed

To solve this problem, we need to understand the relationship between the linear velocity of the center of mass \(v_{\mathrm{cm}}\) and the angular velocity \(\omega\). The standard equation relating these is \(v_{\mathrm{cm}} = R \omega\), which holds true when the wheel rolls without slipping and moves a distance equal to one full circumference of the wheel \(2 \pi R\).
03

Condition analysis

Since the wheel moves a distance shorter than \(2 \pi R\) in one rotation, it cannot be rolling without slipping. This means that the actual distance covered (linear velocity) in one rotation is less than what would be if there were no slipping. Therefore, in this scenario, \(v_{\mathrm{cm}} < R \omega\).
04

Selecting the correct answer

Based on our analysis, option (a) or option (b) describe the correct relationship since both state \(v_{\mathrm{cm}} < R \omega\). Since both are equivalent for our question, we select either. Thus, the correct option is \(v_{\mathrm{cm}} < R \omega\).

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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, often represented by the Greek letter \( \omega \), describes how fast an object rotates or revolves relative to another point, usually the center or a fixed axis. Think of it as the speed at which something spins.
  • It's measured in radians per second, which quantifies angular displacement over time.
  • In our example with the wheel, \( \omega \) tells us how rapidly the wheel turns.
When a wheel completes one full rotation, it covers an angular distance of \( 2\pi \) radians. If the wheel moves a shorter distance than its circumference, it implies a slower angular movement relative to the expected rotation.
Velocity of the Center of Mass
The center of mass velocity, \( v_{\mathrm{cm}} \), is a crucial aspect in analyzing motion. It's the speed of a point which is the average position of all the parts of the system, weighted according to their masses. For a rolling wheel:
  • \( v_{\mathrm{cm}} \) typically equals the radius \( R \) multiplied by angular velocity \( \omega \) when rolling without slipping.
  • However, our wheel covers less than its full circular path, indicating the center of mass velocity is less than \( R \omega \).
Understanding \( v_{\mathrm{cm}} \) aids in predicting how objects move, especially when external forces come into play.
The Dynamics of Rolling Motion
Rolling motion combines two types of motion: translational and rotational. This is the dual motion seen when a wheel rolls over a surface.
  • Translational motion moves the wheel's center of mass linearly from one place to another.
  • Rotational motion involves the turning of the wheel around its center of mass.
For pure rolling (without slipping), these motions perfectly synchronize such that the linear distance covered in one rotation equals the wheel's circumference. However, when the path is shorter, slippage occurs, altering this relationship.
Understanding Slipping in Motion
Slipping happens when there's a disparity between the wheel's angular motion and the actual distance covered. It's a sign of imperfect rolling.
  • In perfect conditions, \( v_{\mathrm{cm}} = R \omega \).
  • However, with slipping, the covering distance becomes lesser, showing \( v_{\mathrm{cm}} < R \omega \).
Slipping can result from insufficient friction or uneven surfaces, leading to less efficient translations of angular velocity to linear motion. Identifying and managing slipping is vital for maintaining the stability and efficiency of rolling bodies in practical applications and theoretical physics.

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Most popular questions from this chapter

An imperfectly rough sphere moves from rest down a plane inclined at an angle \(\alpha\) to the horizontal. The coefficient of friction between the inclined plane and sphere is \(\mu\). Then (a) if \(\mu<\frac{2}{7} \tan \alpha\), then the sphere never rolls (b) if \(\mu=\frac{2}{7} \tan \alpha\), then the maximum friction always being exerted (c) \(\mu>\frac{2}{7} \tan \alpha\), then pure rolling takes place (d) all the above

Mark correct option or options: (a) Rolling friction always oppose the motion of centre of mass of rolling body (b) Sliding friction always oppose the motion of centre of mass of rolling body (c) Rolling friction depends upon hardness of the surface (d) Rolling friction does not depend upon roughness of the surface (e) (a), (c) and (d) are correct

Ram says, "A body may be in pure rotation in the presence of a single external force." Shyam says, "This is possible only for a non rigid body", then: (a) Ram's statement is correct (b) both statements are correct in different situations (c) both statements are wrong (d) both statements are stated by physicists

In a radioactive decay, a number of fragments are found. If parent nucleus is initially at rest then after decay centre of mass will : (a) move on a straight line (b) move in a circle (c) remain in rest (d) move in parabolic path

Mark correct option or options: (a) For neutral equilibrium, the potential energy is constant (b) In stable equilibrium, potential energy is minimum (c) For unstable equilibrium, potential energy is neither constant nor minimum (d) All the above
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