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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 \(a_{x}=r \alpha_{z}\). 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 \(\mu_{\mathrm{k}}\). (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

Draw the Free-body Diagram and Write the Forces

For the cylinder, two forces act vertically, the normal force \(N\) and the weight \(W\). Horizontally, there is only the friction force \(F_{f}\). From Newton's second law, \(N = W = Mg\), and the friction force \(F_{f}= \mu_{k}N = \mu_{k}Mg\). The forces lead us to two equations: \(M a_{x} = F_{f}\) and \(I \alpha_{z} = -R F_{f}\), (\(-R F_{f}\) the negative sign because the friction is acting in the opposite direction to the rotation). Substituting the moments of inertia \(I = 0.5MR^{2}\) into these equations and we get \(0.5 M R^{2} \alpha_{z} = - R \mu_{k}M g\) and \(M a_{x} = \mu_{k} M g\). We can simplify these to get \(a_{x} = \mu_{k}g\) and \(\alpha_{z} = -2 \mu_{k}g / R\).

02

Find the Distance Before Slipping Stops

To find the distance cylinder rolls before slipping stops, we use the second equation for rolling without slipping \(v_{x} = r \omega_{z}\). Also, remember \(v_{x} = a_{x}t\) and \(\omega_{z}=\omega_{0} + \alpha_{z}t\). This gives us \(a_{x}t = R (\omega_{0} + \alpha_{z}t)\). Substituting earlier results gives us the time \(t\) it takes to start rolling without slipping: \(t = \omega_{0}R / (3 \mu_{k}g)\). The distance \(d\) before slipping stops can be calculated from the equation of motion: \(d = 0.5 a_{x} t^{2} = 0 \cdot t + 0.5 \mu_{k}g ( \omega_{0}R / 3 \mu_{k}g)^{2} = \omega_{0}^{2}R^{2} / 6\mu_{k}g\).

03

Calculate the Work Done By the Friction Force

The work done by the friction is calculated using the formula \(W = F_{d} \times d\), where \(F_{d}\) is the friction force and \(d\) is the distance. Substituting the values we've found gives \(W = \mu_{k}Mg (\omega_{0}^{2}R^{2} / 6\mu_{k}g) = \omega_{0}^{2}M R^{2} / 6\).

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Key Concepts

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

Kinetic Friction

Kinetic friction is a force that opposes the relative motion between two surfaces in contact during sliding. It plays a critical role in our daily lives, providing the necessary resistance to enable us to walk or drive without slipping. Let's examine its impact on an object that is rolling without slipping. In this state, while freely rolling, the kinetic friction is negligible. However, the kinetic friction becomes significant when an object, such as a cylinder, starts off by slipping on a surface. It acts in the opposite direction to the motion of the object's center of mass and is proportional to the normal force exerted by the surface on the object, with a coefficient of kinetic friction, \( \mu_k \), governing its magnitude.

In our exercise example, as the cylinder begins to move from rest and slips, the kinetic friction is responsible for both slowing down the relative motion at the point of contact and establishing a condition of rolling without slipping. This frictional force is given by \( F_f = \mu_k N = \mu_k Mg \), where \( N \) is the normal force, equal to the weight \( W \) of the cylinder. The kinetic friction eventually transforms the initial sliding motion into pure rolling, illustrating its dual role in both impeding and aiding motion under different circumstances.

Angular Acceleration

Angular acceleration, \( \alpha_z \), is a measure of how quickly an object's rate of rotation changes with time. In our scenario, it is directly linked to the torque applied by the force of kinetic friction which causes the cylinder to undergo rotational acceleration. One can think of angular acceleration as the rotational counterpart to linear acceleration in translational motion.

The equation \( I \alpha_{z} = -R F_{f} \) from Newton's second law for rotation relates the angular acceleration of the cylinder to the friction force. Here, we observe that the angular acceleration is inversely proportional to the cylinder's moment of inertia and directly proportional to the force of friction and the radius of the cylinder. It is crucial for the cylinder's transition from slipping to rolling without slipping as it directly influences the time and distance it takes the cylinder to achieve a state of steady rolling.

Free-Body Diagram

A free-body diagram is a powerful analytical tool used to illustrate all the forces acting on an object. It simplifies complex physical situations and is essential for solving dynamics problems. The free-body diagram for the cylinder provides a clear visual representation of the forces at play: the weight \( W \), the normal force \( N \) acting upwards, and the kinetic friction force \( F_f \) acting in the horizontal direction, opposite to the direction of slipping or intended motion.

Creating the free-body diagram entails identifying all the forces, their direction, and their points of application. By drawing these forces acting on the body's center of mass, the diagram helps us write down the equations necessary to describe the motion of the cylinder comprehensively. This visualization is crucial to solve for acceleration, the angular acceleration, the distance the cylinder travels before rolling without slipping, and the work done by the friction force.

Moment of Inertia

The moment of inertia, \( I \), quantifies an object's resistance to changes in its rotational motion about an axis. This property depends on the object's mass distribution relative to the axis of rotation. In our cylinder example, the moment of inertia is a crucial factor determining how the angular acceleration and the kinetic friction interact. The moment of inertia for a solid cylinder rotating about an axis through its center is \( I = 0.5 M R^2 \), where \( M \) is the mass of the cylinder and \( R \) is its radius.

The relationship between moment of inertia and angular acceleration is seen through the equation \( I \alpha_{z} = -R F_{f} \), known as the rotational form of Newton's second law. Since the moment of inertia and radius are properties of the cylinder, they play a defining role in how quickly the cylinder can achieve rolling without slipping. A larger moment of inertia would mean a greater resistance to change in rotational motion, implying a potential increase in the time needed to stop slipping and start rolling without slipping.