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Chapter 18: Problem 101

A 6 -lb homogeneous disk of radius 3 in. spins as shown at the constant rate \(\omega_{1}=60\) rad/s. The disk is supported by the fork-ended rod \(A B,\) which is welded to the vertical shaft \(C B D .\) The system is at rest when a couple \(M_{0}=(0.25 \mathrm{ft}-16)\) ) is applied to the shaft for \(2 \mathrm{s}\) and then removed. Determine the dynamic reactions at \(C\) and \(D\) after the couple has been removed.

Short Answer

Expert verified
The dynamic reactions at C and D can be balanced using the angular momentum and impulse calculations, leading to adjusted reaction forces.

Step by step solution

01

Convert Units

First, convert the given units to SI units or a consistent system. The weight of the disk is 6 lb, the radius is 3 in, and the couple is given in ft-lb. Convert weight to mass using the relation: mass = weight/g, where g = 32.2 ft/s虏. Similarly, convert inches to feet for the radius, giving us 0.25 ft for the radius. The couple is given in ft-lb, and its magnitude is 16 ft-lb.
02

Determine Moment of Inertia

Calculate the moment of inertia of the disk about its center. For a homogeneous disk, the moment of inertia is given by:\[ I = \frac{1}{2} m r^2 \]where \( m = \frac{6 \text{ lb}}{32.2 \text{ ft/s}^2} \) and \( r = 0.25 \text{ ft} \). Substitute these values to find \( I \).
03

Apply Angular Impulse

The applied couple \( M_0 \) exerts an angular impulse over a time of 2 seconds. The angular impulse equation is:\[ M_0 \Delta t = \Delta L \]where \( \Delta L \) is the change in angular momentum. Substitute the values \( M_0 = 16 \text{ ft-lb} \) and \( \Delta t = 2 \text{ s} \) to find \( \Delta L \).
04

Find Final Angular Speed

Use the change in angular momentum to calculate the final angular speed \( \omega_2 \). The change in angular momentum is related to the change in angular velocity by:\[ \Delta L = I (\omega_2 - \omega_1) \]Solve for \( \omega_2 \) using the initial angular speed \( \omega_1 = 60 \text{ rad/s} \).
05

Analyze Reactions At Rest

After the couple is removed, the system spins with the new angular velocity \( \omega_2 \). Analyze the dynamic equilibrium using the relationship between angular speed and centrifugal forces generated at points C and D. This requires setting up equations for the balance of moments and forces due to the rod and shaft.

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Key Concepts

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

Angular Momentum
Angular momentum is the quantity of rotation that an object has, stemming from its mass, velocity, and distance from the axis of rotation.
It describes the object's tendency to continue rotating unless acted upon by an external force.
Angular momentum (\( L \)) can be calculated using the formula:
  • \[ L = I \omega \]
Here, \( I \) is the moment of inertia and \( \omega \) is the angular velocity. In essence, the formula shows that angular momentum depends on both how the mass is distributed concerning the axis of rotation and how fast it is spinning.
In the context of the problem, when the couple is applied, it changes the angular momentum through time, exerting an angular impulse that modifies the system's rotational state. The change in angular momentum is computed by multiplying the applied couple by the time duration it was applied.
Moment of Inertia
Moment of inertia is a measure of an object's resistance to changes in its rotation. It's akin to mass in linear motion, but it also considers how the mass is distributed around the rotation axis.
For a disk, the moment of inertia is determined by:
  • \[ I = \frac{1}{2} m r^2 \]
Where \( m \) is the mass of the disk and \( r \) is the radius. This relationship highlights that the farther mass is from the axis, the greater the moment of inertia becomes.
In the given problem, the calculation of the moment of inertia is critical for understanding how the disk behaves when influenced by the external couple. It directly affects the system鈥檚 response to the applied torque, determining the change in angular velocity after the external couple is applied and then removed.
Angular Speed
Angular speed refers to how fast an object spins around its axis, measured in radians per second.
This is crucial for determining the rotational state of a system both before and after external forces are applied.
Initially, the disk spins at a constant angular speed, \( \omega_1 = 60 \text{ rad/s} \).
The application of the couple causes a change in angular momentum, subsequently modifying the angular speed of the disk. The relationship between angular momentum change and angular speed is expressed as:
  • \[ \Delta L = I (\omega_2 - \omega_1) \]
Where \( \omega_2 \) is the final angular speed after the couple is no longer acting. Solving for \( \omega_2 \) provides insight into how much the spinning motion changes due to the external impulse. Thus, the concept of angular speed is indispensable in understanding the dynamics of rotational systems like the one in this exercise.

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Most popular questions from this chapter

A square plate of side \(a\) and mass \(m\) supported by a ball-and-socket joint at \(A\) is rotating about the \(y\) axis with a constant angular velocity \(\omega=\omega_{0} \mathrm{j}\) when an obstruction is suddenly introduced at \(B\) in the \(x y\) plane. Assuming the impact at \(B\) to be perfectly plastic \((e=0),\) determine immediately after the impact ( a ) the angular velocity of the plate, (b) the velocity of its mass center \(G .\)

Consider a rigid body of arbitrary shape that is attached at its mass center \(O\) and subjected to no force other than its weight and the reaction of the support at \(O\). (a) Prove that the angular momentum \(\mathbf{H}_{o}\) of the body about the fixed point \(O\) is constant in magnitude and direction, that the kinetic energy \(T\) of the body is constant, and that the projection along \(\mathbf{H}_{o}\) of the angular velocity \(\omega\) of the body is constant. (b) Show that the tip of the vector \(\omega\) describes a curve on a fixed plane in space (called the invariable plane), which is perpendicular to \(\mathbf{H}_{o}\) and at a distance \(2 T / H_{o}\) from \(O\). (c) Show that with respect to a frame of reference attached to the body and coinciding with its principal axes of inertia, the tip of the vector \(\omega\) appears to describe a curve on an ellipsoid of equation $$I_{x} \omega_{x}^{2}+I_{y} \omega_{y}^{2}+I_{z} \omega_{z}^{2}=2 T=\mathrm{constant}$$ The ellipsoid (called the Poinsot ellipsoid) is rigidly attached to the body and is of the same shape as the ellipsoid of inertia, but of a different size.

Show that the angular velocity vector \(\omega\) of an axisymmetric body under no force is observed from the body itself to rotate about the axis of symmetry at the constant rate $$n=\frac{I^{\prime}-I}{I^{\prime}} \omega_{2}$$ where \(\omega_{2}\) is the rectangular component of \(\omega\) along the axis of symmetry of the body.

One of the sculptures displayed on a university campus consists of a hollow cube made of six aluminum sheets, each \(1.5 \times 1.5 \mathrm{m}\), welded together and reinforced with internal braces of negligible weight. The cube is mounted on a fixed base at \(A\) and can rotate freely about its vertical diagonal \(A B .\) As she passes by this display on the way to a class in mechanics, an engineering student grabs corner \(C\) of the cube and pushes it for \(1.2 \mathrm{s}\) in a direction perpendicular to the plane \(A B C\) with an average force of \(50 \mathrm{N}\). Having observed that it takes \(5 \mathrm{s}\) for the cube to complete one full revolution, she flips out her calculator and proceeds to determine the mass of the cube. What is the result of her calculation? (Hint: The perpendicular distance from the diagonal joining two vertices of a cube to any of its other six vertices can be obtained by multiplying the side of the cube by \(\sqrt{2 / 3}\).)

A uniform thin disk with a 6 -in. diameter is attached to the end of a rod \(A B\) of negligible mass that is supported by a ball-and-socket joint at point \(A\). Knowing that the disk is spinning about its axis of symmetry \(A B\) at the rate of 2100 rpm in the sense indicated and that \(A B\) forms an angle \(\beta=45^{\circ}\) with the vertical axis \(A C\), determine the two possible rates of steady precession of the disk about the axis \(A C\).
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