/*! 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 94 Collar \(A\) has a mass of \(3 \... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 94

Collar \(A\) has a mass of \(3 \mathrm{kg}\) and is attached to a spring of constant \(1200 \mathrm{N} / \mathrm{m}\) and of undeformed length equal to \(0.5 \mathrm{m}\). The system is set in motion with \(r=0.3 \mathrm{m}, v_{\theta}=2 \mathrm{m} / \mathrm{s}\), and \(v_{r}=0 .\) Neglecting the mass of the rod and the effect of friction, determine \((a)\) the maximum distance between the origin and the collar, \((b)\) the corresponding speed. (Hint: Solve the equation obtained for \(r\) by trial and error.)

Short Answer

Expert verified
The maximum distance is approximately \(0.447\,m\) and the speed is approximately \(1.34\,m/s\).

Step by step solution

01

Define the Energy Principles

Use the conservation of mechanical energy principle, since there's no internal energy lost due to friction. The total mechanical energy in the system is the sum of kinetic energy and potential energy (due to the spring and any changes in gravitational potential which are negligible here).
02

Set Up Initial Conditions

Initially, the collar is at a distance \( r_0 = 0.3 \) m from the origin, has a radial velocity \( v_r = 0 \) m/s, and a tangential velocity \( v_\theta = 2 \) m/s. The initial potential energy due to the spring is \( U_{spring, 0} = \frac{1}{2} k (r_0 - 0.5)^2 \).
03

Calculate Initial Total Energy

The initial kinetic energy \( K_0 = \frac{1}{2} m (v_r^2 + r_0^2 v_\theta^2) \) and the initial potential energy comes from the spring, \( U_{spring, 0} = \frac{1}{2} k (r_0 - 0.5)^2 \). Substitute \( m = 3 \), \( k = 1200 \), \( r_0 = 0.3 \), and \( v_\theta = 2 \) to find these values.
04

Express System's Energy at Maximum Extension

At maximum extension \( r_{max} \), we have \( v_r = 0 \). The total energy will consist of a new kinetic energy \( K_{max} = \frac{1}{2} m r_{max}^2 v_\theta^2 \) plus the potential energy due to the spring \( U_{spring, max} = \frac{1}{2} k (r_{max} - 0.5)^2 \).
05

Set Initial Energy Equal to Maximum Extension Energy

The principle of conservation of energy states that initial total energy equals the total energy at maximum extension: \( K_0 + U_{spring, 0} = K_{max} + U_{spring, max} \). Solve this equation for \( r_{max} \).
06

Solve the Equation for Maximum Extension

Substitute the values into your equation and solve for \( r_{max} \) using trial and error or a numerical method. \( r_{max} \) is found when \( v_r \) goes to zero again.
07

Determine Corresponding Speed

Once \( r_{max} \) is known, use the condition that total energy equals kinetic plus potential energies with \( v_r = 0 \) to calculate \( v_\theta \) at maximum extension. Utilize conservation of angular momentum if needed, expressed as \( m v_\theta r = \text{constant} \).

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Key Concepts

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

Mechanical Energy
In mechanics, mechanical energy is the sum of potential energy and kinetic energy. It represents the conserved total energy of a system when there is no external force doing work on that system. In this particular exercise, we utilize the principle of **conservation of mechanical energy** since the system is frictionless, and energy is only transformed between kinetic and potential forms.
To solve the given exercise, recognize:
  • Total mechanical energy = kinetic energy + potential energy.
  • Energy transformations are key: as the collar moves, the energy switches between potential and kinetic.
This **conservation principle** helps find the maximum stretch of the spring and the corresponding speed of the collar.
Kinetic Energy
Kinetic energy is the energy possessed by a moving object. It depends on the mass and velocity of the object. The formula for kinetic energy is:\[K = \frac{1}{2} m v^2\]where \(m\) is mass and \(v\) is velocity. In this exercise, the collar has both radial and tangential components of velocity. Initially, only the tangential component is non-zero, as given by:
  • Initial tangential velocity, \( v_\theta = 2 \text{ m/s}\).
  • Initial radial velocity, \( v_r = 0 \text{ m/s}\).
The initial kinetic energy combines both components:\[K_0 = \frac{1}{2} m (v_r^2 + r^2 v_\theta^2)\]Understanding this division of kinetic energy is crucial for conserving energy throughout the system's motion.
Potential Energy
Potential energy in this scenario stems primarily from the compression or extension of the spring. It is defined by the following formula for a spring obeying Hooke鈥檚 Law:\[U_{spring} = \frac{1}{2} k (r - l_0)^2\]where \(k\) is the spring constant, \(r\) is the current length of the spring, and \(l_0\) is the natural (undeformed) length. Here, \(k = 1200 \text{ N/m}\), and \( l_0 = 0.5 \text{ m}\).
The role of potential energy is to account for the spring's compression or expansion. As the collar moves away or towards the origin, spring's potential energy increases or decreases accordingly, storing energy that governs the system's mechanical behavior.
Radial Velocity
Radial velocity describes the rate of change of the collar's position relative to the origin along a straight line. In the problem setup, initially, this velocity is zero:
  • Initial radial velocity, \( v_r = 0 \text{ m/s}\).
This initial state hints that initially, all the motion is in tangential direction rather than towards or away from the origin. Later in the system's motion, \( v_r \) may change, but the given problem focuses on finding when it returns to zero at the maximum extension.
Understanding radial velocity's behavior ensures maintaining the conservation principle and correctly tracking energy shifts.
Angular Momentum
Angular momentum is a measure of the rotational motion of the collar. It remains constant when no external torques act on the system, a concept that aligns with the conservation of energy. **Conservation of angular momentum** is expressed as:\[m v_\theta r = \text{constant}\]where \(m\) is the mass, \(v_\theta\) the tangential velocity, and \(r\) the distance from the origin.
In our exercise, initially and at maximum extension:
  • The conservation of angular momentum allows determining changes in \(v_\theta\) as the collar reaches its farthest point.
  • Helps cross-verify results obtained from the conservation of mechanical energy.
Understanding angular momentum is essential to predict rotational dynamics as energy transitions occur.

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