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Chapter 6: Problem 30

When the disc stops skidding and begins to roll without slipping, its speed will be (A) \(R \omega_{0}\) (B) \(\frac{1}{3} R \omega_{0}\) (C) \(\frac{1}{2} R \omega_{0}\) (D) \(\frac{1}{4} R \omega_{0}\)

Short Answer

Expert verified
The speed of the disc when it begins to roll without slipping is \(R \omega_{0}\). So, the answer is (A) \(R \omega_{0}\).

Step by step solution

01

Identify given information

The disc initially is skidding and finally it begins to roll without slipping. The question asks for the final speed of the disc. We need to understand the relationship between speed, radius of the disc and angular velocity in case of rolling without slipping situation.
02

Understand the situation of rolling without slipping

When an object rolls without slipping, the speed of its center of mass is given by the relationship \(v = R \omega\), where \(v\) is the speed, \(R\) is the radius, and \(\omega\) is the angular velocity. In this case, \(\omega_{0}\) appears to be the initial angular velocity of the disc.
03

Apply the rolling without slipping condition

Since the disc is now rolling without slipping, we can set \(v = R \omega_{0}\) to find the disc's speed. Thus, the speed of the disc when it begins to roll without slipping will be \(R \omega_{0}\).

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Key Concepts

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

Angular Velocity
Angular velocity is a term used to describe the rate at which an object rotates or revolves around a point or axis. Think of it as how fast something is spinning. Mathematically, it is often denoted by the Greek letter \(\omega\) and measured in radians per second.
  • Angular velocity (\(\omega\)) differs from linear velocity (\(v\)), as it specifically pertains to how quickly an object is moving through its rotational path.
  • This concept plays a crucial role when discussing objects that roll, such as wheels or discs, since their speed is tied to their angular velocity.
When an object like a disc rolls without slipping, the point of contact with the surface does not slide. Instead, the speed of the center of the mass matches the product of angular velocity and the radius, expressed in the equation \(v = R \omega\). Here, \(R\) represents the radius of the disc, ensuring the rolling motion remains constant.
Center of Mass
The center of mass is an important concept in physics, particularly when dealing with motion. It is the point in an object where the mass is considered to be concentrated for calculations of motion.
  • Think of it as the "balancing point" of an object, around which all rotational motion occurs.
  • When analyzing rolling motions, the speed of the center of mass is crucial in determining the motion of the object as a whole.
Understanding the center of mass helps explain why an object rolls without slipping. In a uniformly dense and symmetrical object like a disc, the center of mass follows a straightforward path, and it moves with a velocity described by \(v = R \omega\) when rolling smoothly. Thus, in the absence of forces causing skidding, the entire disc rolls at this speed.
Kinematics
Kinematics is the branch of physics that deals with the motion of objects, focusing on describing the position, velocity, and acceleration, without considering the forces that cause this motion. It's like telling the story of an object's motion.

In the context of rolling without slipping:
  • It involves analyzing how different aspects of an object's motion relate to one another.
  • Kinematic equations enable us to predict the disc's behavior as it transitions from skidding to rolling without slipping.
The equation \(v = R \omega\) emerges from kinematic principles, linking linear and angular velocity for an object in rolling motion. By applying these concepts, we learn that when a disc rolls without slipping, its center of mass moves predictably, maintaining steady contact with the surface. This key kinematic insight helps resolve problems related to rolling objects efficiently.

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Most popular questions from this chapter

A horizontal disc rotates freely about a vertical axis through its centre. A ring, having the same mass and radius as the disc, is now gently placed on the disc. After some time, the two rotate with a common angular velocity (A) some friction exists between the disc and the ring. (B) the angular momentum of the disc plus ring is conserved. (C) the final common angular velocity is \(\frac{2}{3} \mathrm{rd}\) of the initial angular velocity of the disc. (D) \(\frac{2}{3}\) rd of the initial kinetic energy changes to heat,

A small particle of mass \(m\) is projected at an angle \(\theta\) with the \(x\)-axis with an initial velocity \(v_{0}\) in the \(x-y\) plane as shown in Fig. \(6.45\). At a time \(t<\left(v_{0} \sin \theta / g\right)\), the angular momentum of the particle is (A) \(-m g v_{0} t^{2} \cos \theta \hat{j}\) (B) \(m g v_{0} t \cos \theta \hat{k}\) (C) \(-\frac{1}{2} m g v_{0} t^{2} \cos \theta \hat{k}\) (D) \(\frac{1}{2} m g v_{0} t^{2} \cos \theta \hat{i}\)

A string is wrapped several times round a solid cylinder of mass \(m\) and then the end of the string is held stationary while the cylinder is released from rest with no initial motion. The acceleration of the cylinder will be \(\frac{n g}{3}\), then the value of \(n\) is.

A circular disc \(X\) of radius \(R\) is made from an iron plate of thickness \(t\), and another disc \(Y\) of radius \(4 R\) is made from an iron plate of thickness \(t / 4\). Then the relation between the moment of inertia \(I_{X}\) and \(I_{Y}\) is (A) \(I_{Y}=32 I_{X}\) (B) \(I_{Y}=16 I_{X}\) (C) \(I_{Y}=I_{X}\) (D) \(I_{Y}=64 I_{X}\)

4 A uniform rod of mass \(m\) and length \(l\) is suspended horizontally in equilibrium by two springs of spring constant \(k\) each. First spring is connected at the end \(A\) and the second spring is connected \(1 / 4\) away from the other end \(B\). The second spring is now cut. Just after that the acceleration of centre of mass of the rod is \(a_{c m}\), that of end \(A\) is \(a_{A}\) and angular acceleration \(\alpha\) $$ \begin{array}{ll} \text { Column-I } \text { Column-II } \\ \hline \begin{array}{ll} \text { (A) } 12\left|a_{c m}\right| & \text { (1) } 9 g \\ \text { (B) }\left|a_{A}\right| & \text { (2) } g \\ \text { (C) }\left|\frac{l \alpha}{2}\right| & \text { (3) } \frac{g}{3} \\ \text { (4) } 8 g \end{array} \end{array} $$
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