91Ó°ÊÓ

A \(0.2-\mathrm{kg}\) spool slides down along a smooth rod. If the rod has a constant angular rate of rotation \(\dot{\theta}=2 \mathrm{rad} / \mathrm{s}\) in the vertical plane, show that the equations of motion for the spool are \(\ddot{r}-4 r-9.81 \sin \theta=0\) and \(0.8 \dot{r}+N_{s}-1.962 \cos \theta=0,\) where \(N_{s}\) is the magnitude of the normal force of the rod on the spool. Using the methods of differential equations, it can be shown that the solution of the first of these equations is \(r=C_{1} e^{-2 t}+C_{2} e^{2 t}-(9.81 / 8) \sin 2 t .\) If \(r, \dot{r},\) and \(\theta\) are zero when \(t=0,\) evaluate the constants \(C_{1}\) and \(C_{2}\) to determine \(r\) at the instant \(\theta=\pi / 4 \mathrm{rad}\).

Step by step solution

01

Setting Initial Conditions

From the given, \(r\), \(\dot{r}\) and \(\theta\) are all zero when \(t=0\). In the solution of the differential equation \(r=C_{1} e^{-2 t}+C_{2} e^{2 t}-(9.81 / 8) \sin 2 t \), we substitute \(t = 0\) which would give \(r = C_{1} + C_{2}\). Therefore at \(t=0\), \(C_{1} + C_{2} = 0\). This is the first equation needed to solve for the constants.

02

Differentiating the Solution

To find the second equation involving the constants, we differentiate the expression for \(r\). We get \(\dot{r} = -2C_{1} e^{-2 t} + 2C_{2} e^{2 t} - (9.81 / 4) \cos 2t \). Substituting \(t = 0\) we get \(\dot{r} = -2C_{1} + 2C_{2}\). Given \(\dot{r} = 0\), we have \(2C_{1} = 2C_{2}\), hence \(C_{1} = C_{2}\). This is the second equation.

03

Solving the System of Equations

Solving the system of equations \(C_{1} + C_{2} = 0\) and \(C_{1} = C_{2}\) we find \(C_{1} = C_{2} = 0\). Substituting these values into the solution equation would give the radial distance of the spool at any time \(t\).

04

Evaluating for theta = pi/4 rad

The value of \(r\) at \(\theta = \pi / 4 \mathrm{rad}\) can only be determined if we know the relationship between time and the angle \(\theta\) which is not provided in the question. Without this relationship, the exact calculation of \(r\) cannot be done.

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Key Concepts

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

Differential Equations

Differential equations play a pivotal role in modeling the motion of mechanical systems like the sliding spool in this exercise. They describe how a quantity, such as position, changes over time. Here, we consider how the radial distance, denoted as \( r \), changes as the spool moves outward or inward on the rotating rod. The equation governing this is \[ \ddot{r} - 4r - 9.81 \sin \theta = 0 \]which is a second-order differential equation.

• \( \ddot{r} \): Represents the acceleration, or how quickly the velocity of the spool changes.
• The term \(-4r\): Demonstrates a restoring force, tending to bring \( r \) back to a central position.
• The term \(-9.81 \sin \theta\): Accounts for the gravitational component acting along the spool's motion.

Understanding and solving such equations requires mathematical techniques like integration and differentiation, as they provide essential insights into the physics of the system.

Angular Motion

Angular motion describes how a body moves around a central point or axis, in this case, the rod. The spool rotates around this central point with an angular velocity of \( \dot{\theta} = 2 \text{ rad/s} \). Angular motion is characterized by several key properties:

• Angular displacement \(\theta\): Measures how far the object has rotated.
• Angular velocity \(\dot{\theta}\): Describes the rate of change of \(\theta\), indicating how fast the rotation is.
• Angular acceleration \(\ddot{\theta}\): Might describe changes in rotational speed over time, though it's constant here.

In dynamics, angular motion influences forces and torques on objects. Even a smooth rod's constant rotation affects the spool's motion, as we see in the differential equations derived for \(r\).

Initial Conditions

Initial conditions are crucial for solving differential equations because they help determine the specific solution that fits the physical situation. In this problem, the initial conditions are given as:

• \(r = 0\), \(\dot{r} = 0\), and \(\theta = 0\) when \(t = 0\).

These conditions imply that at the starting time, the spool is at the origin, moving neither toward nor away from the center. By substituting these conditions into the general solution:\[ r = C_1 e^{-2t} + C_2 e^{2t} - \frac{9.81}{8} \sin 2t \]We create equations to solve for constants \( C_1 \) and \( C_2 \). They ensure the solution truly reflects the behavior of the system when it starts.

Mechanical Systems

Mechanical systems essentially consist of interconnected parts whose motion can be explained through the principles of physics and dynamics. Here, the mechanical system includes the spool, rod, and gravitational forces acting on them. Mechanical systems are typically analyzed in terms of forces, motion, and energy. These concepts combine to determine how a system behaves over time:

• The interaction between the rod and the spool creates a normal force \( N_s \), affecting the motion equations.
• Gravitational forces play a role in adjusting acceleration, hence appearing in our equation.
• Conservation principles often apply, ensuring energy or momentum doesn't vanish during motion.

By solving equations based on these concepts, one can predict how components within a system will respond to various influences, offering insights into their motion under given conditions.