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Chapter 18: Problem 124

A coin is tossed into the air. It is observed to spin at the rate of 600 rpm about an axis \(G C\) perpendicular to the coin and to precess about the vertical direction \(G D .\) Knowing that \(G C\) forms an angle of \(15^{\circ}\) with \(G D,\) determine \((a)\) the angle that the angular velocity \(\omega\) of the coin forms with \(G D,(b)\) the rate of precession of the coin about \(G D .\)

Short Answer

Expert verified
The angle is 15掳 and the rate of precession is approximately 5.23 rad/s.

Step by step solution

01

Understand the Problem

We are given a coin spinning about an axis perpendicular to the coin, that also precesses about a vertical direction. We need to find two things: the angle of the coin's angular velocity with the vertical and the rate of precession about the vertical direction.
02

Identify Given Information

The coin spins at 600 rpm about axis GC, which is at an angle of 15掳 to the vertical axis GD. We need to translate this angular information into mathematical expressions to solve the problem.
03

Convert RPM to Radians Per Second

Next, convert the spin rate from revolutions per minute to radians per second because angular velocity is usually expressed in radians per unit time. Use the conversion: \[ 600 \text{ rpm} = 600 \times \frac{2\pi}{60} \text{ rad/s} = 20\pi \text{ rad/s}. \]
04

Calculate the Angular Velocity

The angular velocity \(\omega\) is a vector quantity. It will have two components: one along the spin axis GC and another due to the precessional motion along GD. We need the angle it forms with the vertical, GD.
05

Resolve Components of Angular Velocity

The component of angular velocity along GC is the spin, which is \(20\pi\). The angle \(\phi\) of \(\omega\) with GD can be found using trigonometry because \(GC = 15^{\circ}\). The vertical component \(\omega_d\) is given by:\[ \omega_d = 20\pi \cos(15^{\circ}). \]
06

Calculate the Angle 蠅 Forms with GD

The angle \(\theta\) that \(\omega\) forms with GD can be calculated using the tangent relation:\[ \tan(\theta) = \frac{\omega_d \sin(\phi)}{\omega_d \cos(\phi)} = \tan(15^{\circ}). \]Hence, \(\theta = 15^{\circ}. \)
07

Calculate Rate of Precession

Precession rate can be determined by recognizing that it's a rotation about GD axis. The rate \(\Omega\) relates to gyroscopic effect, which can be given (in standard mechanics problems) provided by:\[ \Omega = \frac{T_z}{I \omega}, \]where \(T_z\) is the torque about GD, \(I\) is moment of inertia of the coin, but assuming ideal conditions typically results in:\[ \Omega = \omega \sin(\phi) = 20\pi \sin(15^{\circ}). \]

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Key Concepts

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

Gyroscopic Precession
Gyroscopic precession is a fascinating phenomenon that often appears in physics problems dealing with rotating bodies. When a rotating object experiences a torque, it doesn't simply fall or move in the direction of the force as one might expect; instead, the object rotates at an angle, known as precession.
In the case of the spinning coin, the axis GC is misaligned with the vertical. This misalignment, coupled with the spin, generates a torque that causes the coin to precess about the vertical axis GD.
Key points to consider include:
  • Precession is the result of the interaction between rotational inertia and external torques.
  • The rate of precession depends primarily on the object's angular velocity and the angle of the rotation axis relative to the applied torque.
  • In ideal situations often considered in exercises, simple trigonometric relations help determine precession rates.
Angular Velocity
Angular velocity is a vector quantity describing the rate of rotation around an axis. Whether it's a spinning top, Earth's daily rotation, or a coin tossed in the air, angular velocity provides a measure of how fast an object rotates.
In our solution, the angular velocity about axis GC was first given in rotations per minute (rpm). For practical physics applications, it's often more convenient to express this in radians per second. We did this by converting the spin rate:
\[ 600 \text{ rpm} = 20\pi \text{ rad/s}. \]This conversion helps when expressing angular velocity vectors, as it allows for calculations involving trigonometric functions and other vector components. Here,
  • Angular velocity comprises both spin along the axis GC and precessional motion along GD.
  • The angle of the total angular velocity with a particular axis, like GD, is determined by resolving components using trigonometry.
  • Precession and angular velocity are directly related; understanding this link is crucial to solving complex rotation problems.
Trigonometry in Mechanics
Trigonometry provides powerful tools needed to decipher mechanics problems involving angles and rotational axes. When dealing with rotating bodies, trigonometry helps in breaking down vectors into usable components.
In this exercise, trigonometric functions like cosine and sine were key to resolving angular components along different axes. With the coin:
  • The angle \( \phi = 15^{\circ} \) helped determine necessary component vectors.
  • We used the cosine function to find the vertical component of the angular velocity: \( \omega_d = 20\pi \cos(15^{\circ}) \).
  • Tangent of the angle was used to establish the relationship between different components of vectors: \( \tan(\theta) = \tan(15^{\circ}) \).
By skillfully exploiting trigonometric identities, students can solve for unknown angles and velocities in complex systems, unlocking deeper understanding of motion and forces.

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Most popular questions from this chapter

A space station consists of two sections \(A\) and \(B\) of equal masses that are rigidly connected. Each section is dynamically equivalent to a homogeneous cylinder with a length of \(15 \mathrm{m}\) and a radius of \(3 \mathrm{m}\). Knowing that the station is precessing about the fixed direction \(G D\) at the constant rate of 2 rev/h, determine the rate of spin of the station about its axis of symmetry \(C C^{\prime}\).

The \(85-g\) top shown is supported at the fixed point \(O\). The radii of gyration of the top with respect to its axis of symmetry and with respect to a transverse axis through \(O\) are \(21 \mathrm{mm}\) and \(45 \mathrm{mm}\), respectively. Knowing that \(c=37.5 \mathrm{mm}\) and that the rate of spin of the top about its axis of symmetry is \(1800 \mathrm{rpm},\) determine the two possible rates of steady precession corresponding to \(\theta=30^{\circ} .\)

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.

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 \(A C\).

A solid aluminum sphere of radius 4 in. is welded to the end of a 10 -in-long rod \(A B\) of negligible mass that is supported by a ball- and-socket joint at \(A\). Knowing that the sphere is observed to precess about a vertical axis at the constant rate of \(60 \mathrm{rpm}\) in the sense indicated and that rod \(A B\) forms an angle \(\beta=20^{\circ}\) with the vertical, determine the rate of spin of the sphere about line \(A B .\)
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