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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.

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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.

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Most popular questions from this chapter

A motorcyclist in a circus rides his motorcycle within the confines of the hollow sphere. If the coefficient of static friction between the wheels of the motorcycle and the sphere is \(\mu_{s}=0.4\), determine the minimum speed at which he must travel if he is to ride along the wall when \(\theta=90^{\circ}\). The mass of the motorcycle and rider is \(250 \mathrm{~kg}\), and the radius of curvature to the center of gravity is \(\rho=20 \mathrm{ft}\). Neglect the size of the motorcycle for the calculation. Prob. 13-66

Cartons having a mass of \(5 \mathrm{~kg}\) are required to nove along the assembly line at a constant speed of \(8 \mathrm{~m} / \mathrm{s}\). Determine the smallest radius of curvature, \(\rho\), for the conveyor so the cartons do not slip. The coefficients of static and kinetic friction between a carton and the conveyor are \(u_{s}=0.7\) and \(\mu_{k}=0.5\), respectively. Prob. 13-56

The collar \(A\), having a mass of \(0.75 \mathrm{~kg}\), is attached to a spring having a stiffness of \(k=200 \mathrm{~N} / \mathrm{m}\). When \(\operatorname{rod} B C\) rotates about the vertical axis, the collar slides outward along the smooth \(\operatorname{rod} D E\). If the spring is unstretched when \(s=0\), determine the constant speed of the collar in order that \(s=100 \mathrm{~mm}\). Also, what is the normal force of the rod on the collar? Neglect the size of the collar. Prob. 13-57

Block \(A\) and \(B\) each have a mass \(m\). Determine the largest horizontal force \(\mathbf{P}\) which can be applied to \(B\) so that it will not slide on \(A\). Also, what is the corresponding acceleration? The coefficient of static friction between \(A\) and \(B\) is \(\mu_{s} .\) Neglect any friction between \(A\) and the horizontal surface. Prob. \(13-33\)

If the coefficient of kinetic friction between th \(50-\mathrm{kg}\) crate and the ground is \(\mu_{k}=0.3\), determine th distance the crate travels and its velocity when \(t=3\) : The crate starts from rest, and \(P=200 \mathrm{~N}\).
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