/*! 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 108 The collar, which has a weight o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 108

The collar, which has a weight of 3 lb, slides along the smooth rod lying in the horizontal plane and having the shape of a parabola \(r=4 /(1-\cos \theta)\), where \(\theta\) is in radians and \(r\) is in feet. If the collar's angular rate is constant and equals \(\dot{\theta}=4 \mathrm{rad} / \mathrm{s}\), determine the tangential retarding force \(P\) needed to cause the motion and the normal force that the collar exerts on the rod at the instant \(\theta=90^{\circ}\). Prob. 13-108

Short Answer

Expert verified
The tangential retarding force P and normal force that the collar exerts on the rod at the instant \(\theta=90^{\circ}\) can be calculated using the collar's weight, angular rate, and the parabolic equation of the smooth rod. The actual values depend on the computed acceleration components.

Step by step solution

01

Identify known quantities

The weight of the collar is W = 3 lb. The angular rate, or how fast the angle \(\theta\) changes with time, is constant at \(\dot{\theta}=4 \) rad/s. The equation of the parabolic rod is \(r=4 /(1- \cos \theta)\). The angle at the instant we're interested in is \(\theta=90^{\circ}\).
02

Compute radial and tangential components of acceleration

The collar鈥檚 radial acceleration is \(a_r = r\dot{\theta}^2\). The tangential acceleration is \(a_t = r\ddot{\theta} + 2\dot{r}\dot{\theta}\). But since the rate of change of angle \(\theta\) (denoted as \(\dot{\theta}\)) is constant, \(\ddot{\theta}\) (rate of change of \(\dot{\theta}\)) equals zero, simplifying the equation for \(a_t\) to \(a_t = 2\dot{r}\dot{\theta}\). Calculate the value of \(a_r\), \(\dot{r}\) and \(a_t\) at \(\theta=90^{\circ}\).
03

Determine the retarding force P and normal force that the collar exerts at \(\theta=90^{\circ}\)

The weight W of the collar acts vertically downward while the normal force exerted by the collar is perpendicular to the rod and hence will have radial and tangential components. The normal force equals the collar's normal acceleration times its mass, i.e., N = m * \(a_r\). The tangential retarding force P equals the collar's tangential acceleration times its mass, i.e., P = m * \(a_t\), but directed opposite to \(a_t\). Substituting the computed acceleration components from step 2 and mass (m) based on weight W yields the values for P and N.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Angular Rate
When we talk about angular rate, we refer to how fast an object is rotating or changing its angle per unit of time. In the context of the collar sliding along the smooth rod, angular rate is specifically concerned with how quickly the collar鈥檚 angular position, represented by the variable \( \theta \), is changing. A constant angular rate means that the collar spins or rotates uniformly, without speeding up or slowing down.

The angular rate is expressed as \( \dot{\theta} \) in radians per second, revealing how many radians the collar passes through every second. To visualize this, think of the collar as a point on a spinning wheel. As the wheel spins, the collar traces a circular path, and the rate at which it covers this path is its angular rate.
Radial Acceleration
Radial acceleration is a key concept when you're dealing with circular motion and it's present when an object moves along a curved path, such as our collar on the rod. It's the component of acceleration that points towards the center of the circle along which the object is moving, also known as centripetal acceleration.

Mathematically, the radial acceleration \( a_r \) is given by the formula \( a_r = r\dot{\theta}^2 \), where \( r \) is the radial distance from the center and \( \dot{\theta} \) is the angular rate. It is important to understand that the radial acceleration is what keeps the collar moving along the curved path of the rod, pulling it inward toward the center of curvature of that path.
Normal Force
The concept of normal force is often discussed in the study of dynamics and mechanics, especially when it comes to contact forces. A normal force is the force exerted by a surface perpendicular to the object pressing against it. In the scenario of the collar moving along the rod, the normal force is the component of the force that pushes the collar against the rod.

Since the rod lies in a horizontal plane and has a parabolic shape, the normal force changes direction as the collar moves. However, it always acts perpendicular to the surface of the rod. The normal force is significant because it influences the radial acceleration of the collar. For the collar to continue its path along the rod without losing contact, the normal force must balance the component of the collar's weight that contributes to the radial acceleration.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The conveyor belt delivers each \(12-\mathrm{kg}\) crate to the ramp at \(A\) such that the crate's speed is \(v_{A}=2.5 \mathrm{~m} / \mathrm{s}\), directed down along the ramp. If the coefficient of kinetic friction between each crate and the ramp is \(\mu_{k}=0.3\), determine the speed at which each crate slides off the ramp at \(B\). Assume that no tipping occurs. Take \(\theta=30^{\circ}\).

The pilot of the airplane executes a vertical loop which in part follows the path of a cardioid, \(r=200(1+\cos \theta) \mathrm{m}\), where \(\theta\) is in radians. If his speed at \(A\) is a constant \(v_{p}=85 \mathrm{~m} / \mathrm{s}\), determine the vertical reaction the seat of the plane exerts on the pilot when the plane is at \(A\). He has a mass of \(80 \mathrm{~kg}\). Hint: To determine the time derivatives necessary to calculate the acceleration components \(a_{r}\) and \(a_{\theta}\), take the first and second time derivatives of \(r=200(1+\cos \theta)\). Then, for further information, use Eq. 12-26 to determine \(\dot{\theta}\). Prob. \(13-103\)

The conveyor belt delivers each \(12-\mathrm{kg}\) crate to the ramp at \(A\) such that the crate's speed is \(v_{A}=2.5 \mathrm{~m} / \mathrm{s}\), directed down along the ramp. If the coefficient of kinetic friction between each crate and the ramp is \(\mu_{k}=0.3\), determine the smallest incline \(\theta\) of the ramp so that the crates will slide off and fall into the cart. Probs. \(13-20 / 21\)

A rocket is in free-flight elliptical orbit around the planet Venus. Knowing that the periapsis and apoapsis of the orbit are \(8 \mathrm{Mm}\) and \(26 \mathrm{Mm}\), respectively, determine (a) the speed of the rocket at point \(A^{\prime},(\mathrm{b})\) the required speed it must attain at \(A\) just after braking so that it undergoes an \(8-\mathrm{Mm}\) free-flight circular orbit around Venus, and (c) the periods of both the circular and elliptical orbits. The mass of Venus is \(0.816\) times the mass of the earth. Prob. 13-128

The 4 -kg smooth cylinder is supported by the spring having a stiffness of \(k_{A B}=120 \mathrm{~N} / \mathrm{m}\). Determine the velocity of the cylinder when it moves downward \(s=0.2 \mathrm{~m}\) from its equilibrium position, which is caused by the application of the force \(F=60 \mathrm{~N}\). Prob. 13-34
See all solutions

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Read Explanation

Particle Model of Matter

Read Explanation

Space Physics

Read Explanation

Measurements

Read Explanation

Fundamentals of Physics

Read Explanation

Solid State Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.