91Ó°ÊÓ

When an object is rolling without slipping, the rolling friction force is much less than the friction force when the object is sliding; a silver dollar will roll on its edge much farther than it will slide on its flat side (see Section 5.3\() .\) When an object is rolling without slipping on a horizontal surface, we can approximate the friction force to be zero, so that \(a_{x}\) and \(\alpha_{z}\) are approximately zero and \(v_{x}\) and \(\omega_{z}\) are approximately constant. Rolling without slipping means \(v_{x}=r \omega_{z}\) and \(a_{x}=r \alpha_{x} .\) If an object is set in motion on a surface without these equalities, sliding (kinetic) friction will act on the object as it slips until rolling without slipping is established, A solid cylinder with mass \(M\) and radius \(R\) , rotating with angular speed \(\omega_{0}\) about an axis through its center, is set on a horizontal surface for which the kinetic friction coefficient is \(\mu_{k}\) , (a) Draw a free-body diagram for the cylinder on the surface. Think carefully about the direction of the kinetic friction force on the cylinder. Calculate the accelerations \(a_{x}\) of the center of mass and \(\alpha_{z}\) of rotation about the center of mass. (b) The cylinder is initially slipping completely, so initially \(\omega_{z}=\omega_{0}\) but \(v_{x}=0 .\) Rolling without slipping sets in when \(v_{x}=R \omega_{z}\) . Calculate the distance the cylinder rolls before slipping stops. (c) Calculate the work done by the friction force on the cylinder as it moves from where it was set down to where it begins to roll without slipping.

Step by step solution

01

Free-Body Diagram of the Cylinder

Draw the cylinder with forces acting on it: gravitational force \( Mg \) acting downwards, normal force \( N \) acting upwards, and kinetic friction force \( f_k = \mu_k N \) acting horizontally against the direction of the initial spin. The kinetic friction force acts to oppose the initial sliding motion and spin.

02

Calculate Accelerations

Using Newton's second law and the relationship for rotational motion, we'll find the accelerations. The linear acceleration \( a_x \) is given by \( a_x = \mu_k g \) because the force of friction is \( f_k = \mu_k Mg \). The angular acceleration \( \alpha_z \) is given by \( \alpha_z = \frac{-\mu_k g}{R} \) due to the torque from friction, where \( \tau = I \alpha \) and \( I = \frac{1}{2} MR^2 \).

03

Determine When Rolling Without Slipping Occurs

Rolling without slipping starts when \( v_x = R \omega_z \). We use the equations \( v_x = a_x t = \mu_k g t \) and \( \omega_z = \omega_0 + \alpha_z t = \omega_0 - \frac{\mu_k g}{R} t \) to find \( t \). Solving for \( t \) in \( \mu_k g t = R(\omega_0 - \frac{\mu_k g}{R} t) \) gives \( t = \frac{2 R \omega_0}{3 \mu_k g} \).

04

Calculate the Distance Rolled

The distance rolled during the time \( t \) can be calculated using \( x = \frac{1}{2} a_x t^2 \). Substitute \( a_x = \mu_k g \) and \( t = \frac{2 R \omega_0}{3 \mu_k g} \) into this formula to get \( x = \frac{2 R^2 \omega_0^2}{9 \mu_k g} \).

05

Calculate the Work Done by Friction

Work done by friction \( W = f_k \times x \), where \( f_k = \mu_k Mg \) and \( x = \frac{2 R^2 \omega_0^2}{9 \mu_k g} \). So, substituting values gives: \( W = \mu_k Mg \cdot \frac{2 R^2 \omega_0^2}{9 \mu_k g} = \frac{2}{9} MR^2 \omega_0^2 \).

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Key Concepts

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

Friction

Friction is a force that resists the movement of two surfaces sliding past one another. In the context of rolling motion, such as a cylinder on a horizontal surface, friction plays a crucial role in determining whether the object slides or rolls.
There are two main types of friction involved: static friction and kinetic friction. Static friction occurs when the object is not moving relative to the surface, while kinetic friction occurs when there is sliding motion.
For the cylinder, kinetic friction (\(f_k = \mu_k N = \mu_k Mg\)) acts to oppose its sliding motion. The coefficient of kinetic friction (\(\mu_k\)) is a measure of how resistant a surface is to sliding. This frictional force is responsible for eventually enabling the cylinder to roll without slipping because it works against the relative motion between the cylinder and the surface.

• Static friction leads to rolling without slipping.
• Kinetic friction acts during slipping and is often less than static friction.

As the cylinder transitions from slipping to rolling, the friction begins converting the sliding motion into rotational motion, helping match the linear and angular velocities.

Rotational Dynamics

Rotational dynamics explains the motion of objects as they rotate, considering torque and rotational inertia. For our cylinder, it is essential to understand how the frictional force impacts its rotation.
This circular motion is governed by the force of kinetic friction which creates a torque (\(\tau\)) that alters the cylinder's angular velocity (\(\omega_z\)). Torque is calculated by \(\tau = f_k imes R\), where \(R\) is the radius of the cylinder. This torque results in angular acceleration \(\alpha_z\), which changes the rotational speed over time.
The relationship between torque and angular acceleration is given by \(\tau = I \alpha\), where \(I\) is the moment of inertia. For a solid cylinder, \(I = \frac{1}{2} MR^2\). Therefore, the angular acceleration \(\alpha_z = \frac{-\mu_k g}{R}\).

• The direction of angular acceleration opposes initial spin due to the frictional torque.
• Angular acceleration impacts how quickly the object transitions to rolling without slipping.

Understanding these dynamics is key to predicting future motion states of rotative bodies.

Free-Body Diagram

A free-body diagram is a simple yet powerful tool used to visualize the forces acting on an object. In this context, it helps us examine the horizontal rolling cylinder.
To construct the free-body diagram for the cylinder:

• Gravitational Force (\(Mg\)): Acts downwards towards the center of the Earth.
• Normal Force (\(N\)): Exerted by the ground upwards, balancing out gravity.
• Kinetic Friction Force (\(f_k\)): Acts opposite to the direction of movement, horizontally along the surface.

The kinetic friction force is crucial as it works to stop the sliding motion and initiate rolling without slipping. Understanding how each force acts and their directions is important for setting up equations to describe the motion, such as determining acceleration and final states.

Newton's Second Law

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (\(F = ma\)). It is fundamental for calculating both linear and angular accelerations in rolling motion problems.
For the linear acceleration \(a_x\) of the cylinder, we apply this principle to the friction force: \(a_x = \mu_k g\), as the friction is the only horizontal force acting. Similarly, for angular acceleration \(\alpha_z\), use the torque relation \(\tau = I \alpha\) and friction: \(\alpha_z = \frac{-\mu_k g}{R}\).
Newton's Second Law allows us to link the forces involved with the resulting motions. Both linear and angular calculations depend on \(\mu_k\), the coefficient of kinetic friction, which influences the extent of the accelerations.

• The law helps calculate how quickly the object speeds up or slows down.
• Both linear and rotational equations arise from the same fundamental law stating balance and motion interactions.

This simplified formula application is pivotal for accurately describing the rolling without slipping transition.