/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 109 The ball \(B\) has mass of \(10 ... [FREE SOLUTION] | 91影视

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Chapter 15: Problem 109

The ball \(B\) has mass of \(10 \mathrm{~kg}\) and is attached to the end of a rod whose mass may be neglected. If the rod is subjected to a torque \(M=\left(3 t^{2}+5 t+2\right) \mathrm{N} \cdot \mathrm{m},\) where \(t\) is in seconds, determine the speed of the ball when \(t=2 \mathrm{~s}\). The ball has a speed \(v=2 \mathrm{~m} / \mathrm{s}\) when \(t=0\).

Short Answer

Expert verified
After following these steps, you will be able to find the speed of the ball at \(t = 2\) s.

Step by step solution

01

Find the angular acceleration

Firstly, we need to find the angular acceleration. The torque \(M\) and angular acceleration \(\alpha\) are related by the equation \(M = I\alpha\), where \(I\) is the moment of inertia. In this case, since the mass of the rod is neglected, the ball is essentially a point mass and has a moment of inertia \(I = mR^2\), where \(m\) is the mass of the ball and \(R\) is the distance from the rotation axis, which is the length of the rod. The problem doesn't give us an explicit value for \(R\), but we don't need it because we're looking for the linear velocity of the ball rather than its angular velocity. Therefore, the angular acceleration is \(\alpha = M / I = (3t^2 + 5t + 2) / m\).
02

Find the angular velocity

After finding the angular acceleration, now we will find the angular velocity at a certain time. As the angular acceleration is the derivative of the angular velocity with respect to time, we can find the angular velocity by integrating the angular acceleration function with respect to time. Implementing it, we get \(\omega = \int \alpha dt = \int (3t^2 + 5t + 2) / m dt = (t^3 + \frac{5}{2}t^2 + 2t) / m + C\), where \(C\) is the constant of integration. Put \(t = 0\) and \(\omega = v / R\), where \(v = 2\) m/s as given, to find the value of \(C\).
03

Find the linear velocity

After finding the angular velocity function, now we will find the linear velocity of the ball at \(t = 2\) s. Use the relationship between angular and linear velocities, \(v = \omega R\). Plug in \(t = 2\) into the angular velocity function found to get the angular velocity at this time, and multiply it by \(R\) to get the linear velocity. Note that this assumes that the angle the rod makes with the vertical doesn't change significantly, i.e., small angle approximation is used.

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Key Concepts

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

Moment of Inertia
Moment of inertia, often symbolized by the letter 'I', is a measure of an object's resistance to changes in its rotation rate. It is an analog to mass in linear motion, but instead of simply a mass, it involves the mass and its distribution with respect to the axis of rotation.
Angular Velocity
Angular velocity, symbolized as \( \omega \) is a vector quantity that represents the rate of change of an object's angular position with respect to time. It tells us how fast an object is spinning or rotating. The magnitude of angular velocity is expressed in radians per second (rad/s), and it gives us an idea of how quickly the object is completing its rotations.
Linear Velocity
Linear velocity is a measure of the distance traveled by a point on a rotating object in a specific direction per unit of time. It is directly related to angular velocity through the radius of the rotation path. The relationship can be described by the formula \( v = \omega R \), where \( v \) is the linear velocity, \( \omega \) is the angular velocity, and 'R' is the radius of the circular path.

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Most popular questions from this chapter

A man kicks the 150 -g ball such that it leaves the ground at an angle of \(60^{\circ}\) and strikes the ground at the same elevation a distance of \(12 \mathrm{~m}\) away. Determine the impulse of his foot on the ball at \(A\). Neglect the impulse caused by the ball's weight while it's being kicked.

A ball is thrown onto a rough floor at an angle of \(\theta=45^{\circ} .\) If it rebounds at the same angle \(\phi=45^{\circ},\) determine the coefficient of kinetic friction between the floor and the ball. The coefficient of restitution is \(e=0.6 .\) Hint: Show that during impact, the average impulses in the \(x\) and \(y\) directions are related by \(I_{x}=\mu I_{y}\). Since the time of impact is the same, \(F_{x} \Delta t=\mu F_{y} \Delta t\) or \(F_{x}=\mu F_{y}\).

The truck has a mass of \(50 \mathrm{Mg}\) when empty. When it is unloading \(5 \mathrm{~m}^{3}\) of sand at a constant rate of \(0.8 \mathrm{~m}^{3} / \mathrm{s}\), the sand flows out the back at a speed of \(7 \mathrm{~m} / \mathrm{s}\), measured relative to the truck, in the direction shown. If the truck is free to roll, determine its initial acceleration just as the load begins to empty. Neglect the mass of the wheels and any frictional resistance to motion. The density of sand is \(\rho_{s}=1520 \mathrm{~kg} / \mathrm{m}^{3}\)

A commercial jet aircraft has a mass of \(150 \mathrm{Mg}\) and is cruising at a constant speed of \(850 \mathrm{~km} / \mathrm{h}\) in level flight \(\left(\theta=0^{\circ}\right) .\) If each of the two engines draws in air at a rate of \(1000 \mathrm{~kg} / \mathrm{s}\) and ejects it with a velocity of \(900 \mathrm{~m} / \mathrm{s}\), relative to the aircraft, determine the maximum angle of inclination \(\theta\) at which the aircraft can fly with a constant speed of \(750 \mathrm{~km} / \mathrm{h}\). Assume that air resistance (drag) is proportional to the square of the speed, that is, \(F_{D}=c v^{2}\), where \(c\) is a constant to be determined. The engines are operating with the same power in both cases. Neglect the amount of fuel consumed.

Disk \(A\) has a mass of \(250 \mathrm{~g}\) and is sliding on a smooth horizontal surface with an initial velocity \(\left(v_{A}\right)_{1}=2 \mathrm{~m} / \mathrm{s}\). It makes a direct collision with disk \(B\), which has a mass of \(175 \mathrm{~g}\) and is originally at rest. If both disks are of the same size and the collision is perfectly elastic \((e=1)\), determine the velocity of each disk just after collision. Show that the kinetic energy of the disks before and after collision is the same.
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