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Chapter 10: Problem 54

A uniform solid disk made of wood is horizontal and rotates freely about a vertical axle at its center. The disk has radius \(0.600 \mathrm{~m}\) and mass \(1.60 \mathrm{~kg}\) and is initially at rest. A bullet with mass \(0.0200 \mathrm{~kg}\) is fired horizontally at the disk, strikes the rim of the disk at a point perpendicular to the radius of the disk, and becomes embedded in its rim, a distance of \(0.600 \mathrm{~m}\) from the axle. After being struck by the bullet, the disk rotates at \(4.00 \mathrm{rad} / \mathrm{s}\). What is the horizontal velocity of the bullet just before it strikes the disk?

Short Answer

Expert verified
The horizontal velocity of the bullet just before it strikes the disk can be found by conserving angular momentum and solving for \(v\).

Step by step solution

01

Understand the principle of conservation of angular momentum

Angular momentum, denoted by \(L\), for a rotating object (in this case the bullet-disk system), is given by the product of the moment of inertia (\(I\)) and the angular velocity (\(\omega\)). The conservation of angular momentum states that the total angular momentum of a system remains constant if no external torques act on it. Here, the bullet-disk system is isolated, hence no external torques are acting on it. Hence, the total angular momentum before the bullet hit the disk should be equal to the total angular momentum after the bullet is embedded in the disk and it starts to rotate.
02

Calculate the initial and final angular momenta

Before the collision, the disk is at rest so its angular momentum is zero. The bullet has some initial velocity, say \(v\) (which we have to find), and the initial angular momentum is given by \(L_{initial} = m*v*r\) where \(m=0.020 \, kg\) is the mass of the bullet and \(r=0.600 \, m\) is the radius of the disk. After the collision, the bullet is embedded in the disk and the disk starts to rotate with an angular velocity \( \omega = 4.00 \, rad/s \). The moment of inertia of a disk rotating about a central axis is given by \(I_{disk} = 0.5 * M * R^2\) and for a point mass (bullet) rotating about the center of the disk is given by \(I_{bullet} = m * r^2 \), where \(M = 1.60 \, kg\) is the mass of the disk and \(R=0.600 \, m\) is the disk's radius. The total angular momentum after the collision can be given as \(L_{final} = I_{total} * \omega\), where \(I_{total} = I_{disk} + I_{bullet}\).
03

Equate the initial and final angular momenta and solve for \(v\)

Equate the initial angular momentum (\(m*v*r\)) with the final angular momentum (\(I_{total} * \omega\)) and solve for \(v\). This gives us the initial velocity of the bullet just before it strikes the disk.

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Key Concepts

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

Moment of Inertia
The moment of inertia, often symbolized as \(I\), is a measure of an object's resistance to changes in its angular motion. It acts as the rotational analog to mass in linear motion. The moment of inertia depends on the mass of the object and how that mass is distributed relative to the axis of rotation. For different shapes and mass distributions, the formula for the moment of inertia changes slightly.
For example, in the context of a solid disk rotating about its central axis, the moment of inertia is given by \(I_{disk} = \frac{1}{2} M R^2\), where \(M\) is the mass of the disk and \(R\) its radius. For point masses, like the bullet embedded in the disk, the moment of inertia can be calculated as \(I_{bullet} = m r^2\), where \(m\) is the mass of the bullet and \(r\) the distance from the rotation axis.
  • The total moment of inertia in the disk-bullet system is the sum of the disk's and the bullet's individual moments.
  • This table helps in understanding how the mass distribution affects rotation.
Calculating the total moment of inertia helps in analyzing the system's response to applied forces or torques, crucial for understanding and solving rotational dynamics problems.
Angular Velocity
Angular velocity, denoted by \(\omega\), refers to the rate of change of angular displacement and is a measure of how quickly an object rotates or revolves around an axis. It is typically measured in radians per second (rad/s).
In rotational motion, angular velocity plays a key role analogous to velocity in linear motion. When examining systems like the rotating disk, it's essential to understand how changes in angular velocity affect the dynamics of the system.
  • Angular velocity can be influenced by external forces, torques, or changes in the distribution of mass.
  • When a bullet impacts the disk, the angular velocity is a critical factor in determining the post-collision state of the system.
In the given exercise, the disk, initially at rest, gains an angular velocity of \(4.00 \text{ rad/s}\) after the bullet becomes embedded. This change showcases how the conservation of angular momentum dictates the system's final rotational state.
Rotational Dynamics
Rotational dynamics is the study of objects in rotational motion and involves concepts like torque, angular acceleration, and angular momentum. It translates the principles of linear dynamics to rotational motion and helps in understanding how and why objects rotate. The conservation of angular momentum is a core principle in rotational dynamics.
According to the conservation of angular momentum, the total angular momentum of an isolated system remains constant unless acted upon by external torques. This principle helps in solving problems like the bullet-disk system by equating initial and final angular momentum to find unknown quantities, like the bullet's initial velocity.
  • Angular momentum \(L\) is given by \(L = I \omega\), tying together moment of inertia and angular velocity.
  • By keeping track of angular momentum before and after interactions, one can accurately predict outcomes in rotational systems.
This principle allows us to understand rotational interactions similar to how Newton's laws describe linear interactions, forming the bedrock of rotational dynamics problems.

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Most popular questions from this chapter

A hollow, thin-walled sphere of mass \(12.0 \mathrm{~kg}\) and diameter \(48.0 \mathrm{~cm}\) is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by \(\theta(t)=A t^{2}+B t^{4},\) where \(A\) has numerical value 1.50 and \(B\) has numerical value \(1.10 .\) (a) What are the units of the constants \(A\) and \(B ?\) (b) At the time \(3.00 \mathrm{~s}\), find (i) the angular momentum of the sphere and (ii) the net torque on the sphere.

A large turntable with radius \(6.00 \mathrm{~m}\) rotates about a fixed vertical axis, making one revolution in \(8.00 \mathrm{~s}\). The moment of inertia of the turntable about this axis is \(1200 \mathrm{~kg} \cdot \mathrm{m}^{2}\). You stand, barefooted, at the rim of the turntable and very slowly walk toward the center, along a radial line painted on the surface of the turntable. Your mass is \(70.0 \mathrm{~kg}\). since the radius of the turntable is large, it is a good approximation to treat yourself as a point mass. Assume that you can maintain your balance by adjusting the positions of your feet. You find that you can reach a point \(3.00 \mathrm{~m}\) from the center of the turntable before your feet begin to slip. What is the coefficient of static friction between the bottoms of your feet and the surface of the turntable?

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.

A diver comes off a board with arms straight up and legs straight down, giving her a moment of inertia about her rotation axis of \(18 \mathrm{~kg} \cdot \mathrm{m}^{2}\). She then tucks into a small ball, decreasing this moment of inertia to \(3.6 \mathrm{~kg} \cdot \mathrm{m}^{2}\). While tucked, she makes two complete revolutions in \(1.0 \mathrm{~s}\). If she hadn't tucked at all, how many revolutions would she have made in the \(1.5 \mathrm{~s}\) from board to water?

A solid cylinder with radius \(0.140 \mathrm{~m}\) is mounted on a frictionless, stationary axle that lies along the cylinder axis. The cylinder is initially at rest. Then starting at \(t=0\) a constant horizontal force of \(3.00 \mathrm{~N}\) is applied tangential to the surface of the cylinder. You measure the angular displacement \(\theta-\theta_{0}\) of the cylinder as a function of the time \(t\) since the force was first applied. When you plot \(\theta-\theta_{0}\) (in radians) as a function of \(t^{2}\left(\right.\) in \(\left.\mathrm{s}^{2}\right),\) your data lie close to a straight line. If the slope of this line is \(16.0 \mathrm{rad} / \mathrm{s}^{2},\) what is the moment of inertia of the cylinder for rotation about the axle?
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