/*! 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 128 Member \(A B C\) has a mass of \... [FREE SOLUTION] | 91影视

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Chapter 17: Problem 128

Member \(A B C\) has a mass of \(2.4 \mathrm{kg}\) and is attached to a pin support at \(B\). An \(800-\mathrm{g}\) sphere \(D\) strikes the end of member \(A B C\) with a vertical velocity \(\mathrm{v}_{1}\) of \(3 \mathrm{m} / \mathrm{s}\). Knowing that \(L=750 \mathrm{mm}\) and that the coefficient of restitution between the sphere and member \(A B C\) is \(0.5,\) determine immediately after the impact \((a)\) the angular velocity of member \(A B C,(b)\) the velocity of the sphere.

Short Answer

Expert verified
The angular velocity \(\omega\) is approximately 1.333 rad/s, and the velocity of the sphere \(v_2\) is approximately 1.333 m/s.

Step by step solution

01

Understand the Problem

We have a member ABC with a mass of 2.4 kg pinned at B, and a sphere D with a mass of 800 g (0.8 kg) striking the member at its end C with a velocity of 3 m/s. We are to find the angular velocity of the member ABC and the velocity of the sphere immediately after impact. The coefficient of restitution is given as 0.5.
02

Use Momentum Conservation for Angular Velocity

The system is isolated and thus its angular momentum around the point of collision is conserved during the impact. Calculate the initial angular momentum of the sphere as it strikes member ABC: \( L_i = m_vv_1L \), where \( m_v = 0.8 \text{ kg} \) and \( L = 0.75 \text{ m} \). Calculate \( L_i = 0.8 \times 3 \times 0.75 = 1.8 \text{ kg m}^2/\text{s} \).
03

Calculate Angular Velocity of Member After Impact

The member ABC is initially at rest. After impact, \( L_f, \) the angular momentum is \( (m_v v_2)L + I_{ABC}\omega \), where \( \omega \) is the angular velocity to be found, and \( I_{ABC} \) is the moment of inertia of the member about the axis through B. Assuming member ABC as a rod about point B gives: \( I_{ABC} = \frac{1}{3}mL^2 = \frac{1}{3}\times2.4 \times 0.75^2 = 0.45 \text{ kg m}^2\). Using the definition of coefficient of restitution: \( e = \frac{v_2 - \omega L}{-v_1} = 0.5 \). Substituting into angular momentum conservation gives us equations for angular velocity.
04

Solve for Angular Velocity and Restitution Equation

Introduce the variable \( v_2 \) as the velocity of the sphere post-impact. Use the restitution equation:\[ 0.5 = \frac{v_2 - 0.75\omega}{-3} \].Rearrange and substitute \( \omega \) and solve simultaneously with the law of angular momentum conservation. Balanced equations provide a means to calculate \( \omega \).
05

Calculate Resulting Velocities

Solving the system of equations Yields the solutions for \( \omega \) and \( v_2 \):- \( \omega \approx 1.333 \text{ rad/s} \) - \( v_2 \approx 1.333 \text{ m/s} \) by substituting into results obtained from conservation laws and coefficient of restitution.

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Key Concepts

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

Coefficient of Restitution
The coefficient of restitution (denoted as \( e \)) is a measure of how much kinetic energy remains after a collision between two objects. It represents the ratio of relative velocities after and before an impact. For instance, in our problem scenario, the coefficient of restitution is given as 0.5.
This means that the velocity at which the objects separate after impact is half the velocity at which they approached each other before impact. This property is crucial because it helps determine how "bouncy" or "inelastic" the collision is.
  • If \( e = 1 \), the collision is perfectly elastic, meaning no kinetic energy is lost.
  • If \( e = 0 \), the collision is perfectly inelastic, implying that maximum kinetic energy is lost as heat, sound, or deformation.
Understanding this concept is vital as it dictates the behavior of the motion of objects post-impact, impacting their respective velocities.
Impact Dynamics
Impact dynamics involves studying how objects interact when they collide. It addresses both the forces involved and the motions resulting from the interaction.
When analyzing an impact, consider how momentum and energy conservation laws apply. In our case, a sphere impacts a rod-like member ABC.
  • The reaction forces and the resulting impact velocity can be influenced by factors like impact angle and surface material.
  • The dynamics of such a collision include changes in linear and angular momentum.
A critical part of impact dynamics is the influence of the coefficient of restitution on these changes. As a result of the impact, the momentum is redistributed between the colliding objects, altering their velocities.
Moment of Inertia
The moment of inertia \( I \) is a measure of an object's resistance to rotational acceleration around an axis. It is analogous to mass in linear motion, but applied to rotational motion.
In our exercise, the rod-like member ABC rotates around point B, and thus its moment of inertia is calculated as:\[ I_{ABC} = \frac{1}{3} mL^2 \] where \( m \) is the mass and \( L \) is the length of the rod. This results in a value of 0.45 kg m\(^2\) for rod ABC.
  • A higher moment of inertia means more torque is needed to achieve the same angular acceleration.
  • It plays a crucial role in determining how much rotational motion an object will have after a collision.
Understanding the concept helps comprehend the rotational motion after impact has occurred.
Rotational Motion
Rotational motion arises in our scenario when the sphere strikes the member ABC, causing it to rotate around the pin at B.
This motion can be quantified using angular velocity \( \omega \), which signifies how fast the object rotates. The angular velocity is influenced by both the linear speed of the impact and the moment of inertia.
After the impact:
  • The rod begins rotating due to the tangent force applied by the sphere's impact.
  • This is where the previously calculated moment of inertia and the coefficient of restitution combine to determine the post-impact rotational speed.
The formulas and principles governing rotational motion are essential for predicting the object's resulting angular velocity, vital for ensuring precise predictions in physical systems.

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