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Chapter 17: Problem 71

A cord is wrapped around the outer surface of the 8 -kg disk. If a force of \(F=\left(1 / 4 \theta^{2}\right) \mathrm{N}\), where \(\theta\) is in radians, is applied to the cord, determine the disk's angular acceleration when it has turned 5 revolutions. The disk has an initial angular velocity of \(\omega_{0}=1 \mathrm{rad} / \mathrm{s}\).

Short Answer

Expert verified
Since the disk radius and dimension was not given, it's impossible to provide a numerical value for the angular acceleration. However, the procedure to find the answer is provided above. It involves calculating the net force, the corresponding torque, and the angular acceleration using the formula \(\alpha=\tau / I\).

Step by step solution

01

Calculation of Net Force

Since the applied force is in radians (\(F=\left(1 / 4 \theta^{2}\right) \mathrm{N}\)), we must first convert the 5 revolutions to radians. Remember 1 revolution is \(2\pi\) radians, so 5 revolutions = \(5 *2\pi =10\pi\) radians. Then, substitute \(\theta=10\pi\) into the expression for \(F\), we get: \(F=(1 / 4 *(10\pi)^{2}) \mathrm{N}\).
02

Calculation of Torque

After obtaining the net force, calculate the net Torque using the formula \(\tau=rf\) (Torque equals radius times force). It's worth noting that since the force is being applied at the outer edge of the disk, the radius becomes the radius of the disk (r). Therefore, one must know the radius and must calculate it if it is not given. In this case, since the radius isn't provided, you may substitute with \(r\). Thus, \(\tau=rF\). This gives us the net torque.
03

Calculation of Angular Acceleration

With the net torque, calculate the disk's angular acceleration using the formula \(\alpha=\tau / I\) (angular acceleration equals torque divided by moment of inertia). We can recall from physics that the moment of inertia (I) of a disk is \(I=1 / 2 mr^{2}\). Substituting the net torque and moment of inertia, we will get the disk's angular acceleration when it has turned 5 revolutions.

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Key Concepts

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

Understanding Moment of Inertia
The moment of inertia is a concept similar to mass in linear motion but applied to rotational motion. It's crucial for understanding how different objects rotate. In simple terms, the moment of inertia determines how much an object resists changes in its rotational motion.

For a disk, like the one described in the exercise, the moment of inertia (\(I\)) can be calculated using the formula \(I = \frac{1}{2} mr^2\), where \(m\) is the mass of the disk, and \(r\) is its radius.

  • The higher the moment of inertia, the harder it is to change its rotation.
  • It's affected by both the mass and the distribution of that mass relative to the axis of rotation.
To calculate the angular acceleration, you'll need this value because it plays a crucial role in defining how torque affects rotational motion.
Torque Calculation and Its Importance
Torque is a force that causes objects to rotate. In this exercise, torque is calculated using the equation \(\tau = rF\), where \(\tau\) is torque, \(r\) is the radius of the disk, and \(F\) is the force applied.

Torque is essentially the rotational equivalent of linear force. When a force is applied at some distance from the axis of rotation, it creates torque resulting in rotational motion.

  • Torque depends not only on the amount of force applied but also on the distance from the point of application to the axis of rotation.
  • This distance is usually the radius in problems involving circular motions, like disks, making torque directly proportional to the radius when considering uniform force.
In practical terms, the larger the torque, the faster the object starts spinning or stops spinning.
Grasping Angular Motion
Angular motion is all about how objects rotate around a fixed point, often described using parameters like angular velocity and angular acceleration.

Here are a few basics of angular motion:
  • Angular velocity (\(\omega\)) is the rate at which an object spins around its axis. It's given in radians per second.
  • Angular acceleration (\(\alpha\)) indicates how quickly the angular velocity is changing. It's figured using \(\alpha = \frac{\tau}{I}\), connecting torque and moment of inertia.
  • The relation between these is crucial: torque directly influences angular acceleration, which in turn affects angular velocity.
Understanding these terms helps to interpret scenarios where objects, like the disk in the exercise, change their rotational speed due to external forces.

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Most popular questions from this chapter

The sphere is formed by revolving the shaded area around the \(x\) axis. Determine the moment of inertia \(I_{x}\) and express the result in terms of the total mass \(m\) of the sphere. The material has a constant density \(\rho\).

Gears \(A\) and \(B\) have a mass of \(50 \mathrm{~kg}\) and \(15 \mathrm{~kg}\), respectively. Their radii of gyration about their respective centers of mass are \(k_{C}=250 \mathrm{~mm}\) and \(k_{D}=150 \mathrm{~mm}\). If a torque of \(M=200\left(1-e^{-0.2 t}\right) \mathrm{N} \cdot \mathrm{m},\) where \(t\) is in seconds, is applied to gear \(A,\) determine the angular velocity of both gears when \(t=3 \mathrm{~s}\), starting from rest.

The solid cylinder has an outer radius \(R\), height \(h\), and is made from a material having a density that varies from its center as \(\rho=k+a r^{2},\) where \(k\) and \(a\) are constants. Determine the mass of the cylinder and its moment of inertia about the \(z\) axis.

The jet aircraft has a total mass of \(22 \mathrm{Mg}\) and a center of mass at \(G .\) Initially at take-off the engines provide a thrust \(2 T=4 \mathrm{kN}\) and \(T^{\prime}=1.5 \mathrm{kN}\). Determine the acceleration of the plane and the normal reactions on the nose wheel at \(A\) and each of the two wing wheels located at \(B\). Neglect the mass of the wheels and, due to low velocity, neglect any lift caused by the wings.

The solid ball of radius \(r\) and mass \(m\) rolls without slipping down the \(60^{\circ}\) trough. Determine its angular acceleration.
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