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Chapter 18: Problem 98

Assuming that the wheel of Prob. 18.97 weighs 8 lb, has a radius \(a=4 \mathrm{in.},\) and a radius of gyration of \(3 \mathrm{in}\), and that \(R=20 \mathrm{in}\), determine the force exerted by the plate on the wheel when \(\Omega=25 \mathrm{rad} / \mathrm{s}\).

Short Answer

Expert verified
The force exerted by the plate on the wheel is approximately 25.93 lb.

Step by step solution

01

Understand the Problem

We need to determine the force exerted by the plate on the wheel. This involves understanding the physical system: a wheel rotating with angular velocity \( \Omega = 25 \text{ rad/s} \), while being acted upon by centrifugal and gravitational forces. The radius of the wheel \( a = 4 \text{ in} \), and the radius of the rotation \( R = 20 \text{ in} \).
02

Convert Units

Let's first convert the weight of the wheel to pounds-force and express all distances in feet:- Weight \( = 8 \text{ lb} \).- Radius of the wheel \( a = \frac{4}{12} \text{ ft} = \frac{1}{3} \text{ ft} \).- Radius of gyration already provided in inches and contributes to rotational dynamics.
03

Calculate Centrifugal Force

Using the centrifugal force formula \( F_c = mR\Omega^2 \), where \( m \) is the mass:- First, convert the weight to mass: \( m = \frac{8}{32.2} \text{ slug} \) (using \( 32.2 \text{ ft/s}^2 \) as gravitational acceleration).- Then, substitute known values: \( F_c = \left( \frac{8}{32.2} \right) \times \left( \frac{20}{12} \right) \times 25^2 \).
04

Solve for Force Exerted by the Plate

Since the force exerted by the plate is the centripetal force required to keep the wheel in rotation:- Solve the equation from Step 3 to find \( F_c \), which is the force exerted by the plate.- Calculate: \( F_c = \frac{8}{32.2} \times \frac{20}{12} \times 625 \approx 25.93 \text{ lb} \).

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Key Concepts

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

Centrifugal Force
Centrifugal force is a concept that often causes confusion. Even though it seems like a force pushing an object outward while it spins, centrifugal force is actually not a real force. Instead, it's a perceived effect due to an object鈥檚 tendency to maintain its straight-line motion because of inertia.
When you swing a ball tied to a string in a circle, you feel it's being pulled outward. But what's happening is the ball wants to continue in a straight line and the string provides the necessary force to hold it in circular motion.
Centrifugal force arises from the rotation perspective. If you're riding in a spinning drum and feel pressed against the wall, that's because your inertia wants you to travel in a straight line and the drum wall pushes inward, providing necessary centripetal force, but you perceive it as an outward force. It鈥檚 important for understanding rotational dynamics as it relates to angular motion.
Angular Velocity
Angular velocity is a measure of how quickly an object rotates around a fixed point. It is vital to understanding how objects behave when they spin. It is usually expressed in units like radians per second (rad/s).
Angular velocity differs from linear velocity as it measures rotation rather than straight-line movement. Think of it like the hands of a clock: they rotate around the clock face, and their angular velocity tells us how quickly they make that motion.
In the exercise, the wheel is described as having an angular velocity of 25 rad/s. This indicates how quickly the wheel spins about its axis, which is crucial for calculating forces like the centripetal force keeping it in motion.
Centripetal Force
Centripetal force is necessary for keeping an object moving in a circular path. Rather than being an actual type of force, centripetal refers to the direction of the force: always pointing towards the center around which the object rotates.
For a rotating wheel like in the exercise, the centripetal force is the force exerted by the plate on the wheel, keeping it from flying off at a tangent due to inertia. The formula for centripetal force is \[ F_c = m imes R \times \Omega^2 \]
Here, \( m \) is the mass of the object, \( R \) is the radius of rotation, and \( \Omega \) is the angular velocity. This force ensures a system remains stable during rotation and is crucial for various real-life applications, from planets in orbit to vehicles turning corners.

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Most popular questions from this chapter

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 homogeneous disk with a radius of 9 in. is welded to a rod \(A G\) with a length of 18 in. and of negligible weight that is connected by a clevis to a vertical shaft \(A B\). The rod and disk can rotate freely about a horizontal axis \(A C,\) and shaft \(A B\) can rotate freely about a vertical axis. Initially, rod \(A G\) is horizontal \(\left(\theta_{0}=90^{\circ}\right)\) and has no angular velocity about \(A C\). Knowing that the smallest value of \(\theta\) in the ensuing motion is \(30^{\circ}\), determine \((a)\) the initial angular velocity of shaft \(A B,(b)\) its maximum angular velocity.

The flywheel of an automobile engine, which is rigidly attached to the crankshaft, is equivalent to a 400 -mm-diameter, \(15-\) mm-thick steel plate. Determine the magnitude of the couple exerted by the flywheel on the horizontal crankshaft as the automobile travels around an unbanked curve of \(200-\) m radius at a speed of \(90 \mathrm{km} / \mathrm{h}\), with the flywheel rotating at \(2700 \mathrm{rpm}\). Assume the automobile to have \((a)\) a rear-wheel drive with the engine mounted longitudinally, \((b)\) a front-wheel drive with the engine mounted transversely. (Density of steel \(=7860 \mathrm{kg} / \mathrm{m}^{3}\) )

A thin homogeneous disk with a mass \(m\) and radius \(r\) is mounted on the horizontal axle \(A B\). The plane of the disk forms an angle of \(\beta=20^{\circ}\) with the vertical. Knowing that the axle rotates with an angular velocity \(\omega,\) determine the angle \(\theta\) formed by the axle and the angular momentum of the disk about \(G .\)

A high-speed photographic record shows that a certain projectile was fired with a horizontal velocity \(\bar{v}\) of \(2000 \mathrm{ft} / \mathrm{s}\) and with its axis of symmetry forming an angle \(\beta=3^{\circ}\) with the horizontal. The rate of spin \(\psi\) of the projectile was \(6000 \mathrm{rpm},\) and the atmospheric drag was equivalent to a force \(\mathrm{D}\) of 25 lb acting at the center of pressure \(C_{P}\) located at a distance \(c=6\) in. from \(G .(a)\) Knowing that the projectile has a weight of 45 lb and a radius of gyration of 2 in. with respect to its axis of symmetry, determine its approximate rate of steady precession. ( \(b\) ) If it is further known that the radius of gyration of the projectile with respect to a transverse axis through \(G\) is 8 in, determine the exact values of the two possible rates of precession.
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