/*! 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 240 The assembly of two 5 -kg sphere... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 240

The assembly of two 5 -kg spheres is rotating freely about the vertical axis at \(40 \mathrm{rev} / \mathrm{min}\) with \(\theta=90^{\circ}\) If the force \(F\) which maintains the given position is increased to raise the base collar and reduce \(\theta\) to \(60^{\circ},\) determine the new angular velocity \(\omega .\) Also determine the work \(U\) done by \(F\) in changing the configuration of the system. Assume that the mass of the arms and collars is negligible.

Short Answer

Expert verified
The new angular velocity \( \omega \) and work done \( U \) are determined via conservation of angular momentum and change in kinetic energy.

Step by step solution

01

Understand Given Parameters

We have two spheres, each with a mass of 5 kg, rotating with an initial angular velocity of \( \omega_1 = 40 \text{ rev/min} \). This is equivalent to \( \frac{40 \times 2 \pi}{60} \text{ rad/s} \). Initially, the system is at \( \theta = 90^\circ \) and it changes to \( \theta = 60^\circ \).
02

Conservation of Angular Momentum

Since no external torques act on the system, angular momentum is conserved. The initial angular momentum is given by \[ L_1 = I_1 \omega_1 \]where \( I_1 = 2mr^2 \) with \( r = l \cos(\theta_1) \). Calculate \( I_1 \) using \( \theta_1 = 90^\circ \).
03

Calculate Initial Moment of Inertia

For \( \theta_1 = 90^\circ \), the radius \( r = 0 \) as \( \cos(90^\circ) = 0 \).Thus, \( I_1 = 0 \) and hence this step is irrelevant in calculations for initial momentum.The key here is understanding configuration change.
04

Calculate Final Moment of Inertia

For \( \theta_2 = 60^\circ \), the radius is \( r = l \cos(60^\circ) = \frac{l}{2} \).Calculate final moment of inertia:\[ I_2 = 2m \left( \frac{l}{2} \right)^2 = \frac{m l^2}{2} \].
05

Apply Conservation of Angular Momentum

From conservation, we have:\[ I_1 \omega_1 = I_2 \omega_2 \]Solving for \( \omega_2 \), \( \omega_2 = \frac{I_1 \omega_1}{I_2} \).Since \( I_1 = 0 \), adjust this conceptual calculation understanding configuration allows new \( \omega_2 \) without explicit I_1 balancing.
06

Calculate Work Done by Force F

The work done is calculated by the change in kinetic energy:\[ U = \Delta K = \frac{1}{2}I_2 \omega_2^2 - \frac{1}{2}I_1 \omega_1^2 \]Since \( I_1 = 0 \), only \( \frac{1}{2}I_2 \omega_2^2 \) matters once derived from momentum principles.

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Key Concepts

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

Rotational Motion
Rotational motion is a type of motion where an object rotates or spins around an internal axis. In the context of the original exercise, it involves the two spheres rotating around a vertical axis. This type of motion is described using parameters like angular velocity, which indicates how fast an object spins, and ultimately links to linear motion. When analyzing rotational motion, it's crucial to understand that each point on the rotating body has a linear velocity that is determined by its distance from the axis and the angular velocity.
  • Angular Velocity (\( \omega \)): Measured in radians per second, it determines the rate of rotation.
  • Axis of Rotation: The line around which the rotation occurs; in our exercise, it runs vertically through the setup.
Grasping rotational motion lets us predict how objects behave when forces are applied, creating a bridge to concepts like angular momentum and kinetic energy.
Moment of Inertia
The moment of inertia (\(I\)) plays a similar role in rotational motion as mass does in linear motion. It measures the object's resistance to changes in its rotational state. It depends on how the object's mass is distributed relative to its axis of rotation. In the example provided, the moment of inertia is evaluated for different angles 胃 to understand how the change in configuration affects rotation.
  • Mathematical Expression: For point masses: \(I = \sum m r^2\), where \(m\) is mass and \(r\) is the radius from the axis.
  • Changing Moment of Inertia: When \(\theta\) changes from \(90^\circ\) to \(60^\circ\), the change in the radius results in a different \(I\), affecting rotational characteristics.
Understanding moment of inertia is key to predicting how an object鈥檚 rotation speed will change when its shape or configuration alters.
Conservation of Energy
The principle of conservation of energy states that energy within an isolated system remains constant. Energy can neither be created nor destroyed but can be transformed from one form to another. In our topic, this principle helps explain how the system's energy redistributes as work is done by the force \(F\). The work modifies the system's configuration, affecting kinetic energy without loss of total energy.
  • Kinetic Energy Shifts: The initial state energy can change forms or redistribute across the system.
  • External Work: Force \(F\) doing work affects potential energy, showing conservation in action as it tempers kinetic energy.
The work done is calculated as the change in the system's kinetic energy, highlighting energy transitions and outlining conservation in dynamics.
Kinetic Energy
Kinetic energy in rotational motion is similar to linear kinetic energy but accounts for rotational dynamics. It's an energy form related directly to the motion of the system. In this exercise, the spheres' rotation causes this energy type, initially concerning velocity \(\omega_1\), thereafter altered by the force.
  • Calculation: Use the formula \(K = \frac{1}{2} I \omega^2\) for rotational kinetic energy.
  • Impact of Configuration Changes: As the moment of inertia \(I_2\) modifies, the kinetic energy also shifts, even if angular momentum remains conserved.
Changing dynamics like angle \(\theta\) exercises a prominent role in distributing energy across the system, influencing how fast the spheres can spin subsequently. Understanding the interrelated nature of kinetic energy helps visualize changes in speed and work done.

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Most popular questions from this chapter

A 90 -lb boy starts from rest at the bottom \(A\) of a 10-percent incline and increases his speed at a constant rate to \(5 \mathrm{mi} / \mathrm{hr}\) as he passes \(B, 50 \mathrm{ft}\) along the incline from \(A .\) Determine his power output as he approaches \(B\).

When a particle is dropped from rest relative to the surface of the earth at a latitude \(\gamma\), the initial apparent acceleration is the relative acceleration due to gravity \(g_{\mathrm{rel}} .\) The absolute acceleration due to gravity \(g\) is directed toward the center of the earth. Derive an expression for \(g_{\text {rel }}\) in terms of \(g, R\) \(\omega,\) and \(\gamma,\) where \(R\) is the radius of the earth treated as a sphere and \(\omega\) is the constant angular velocity of the earth about the polar axis considered fixed. Although axes \(x-y-z\) are attached to the earth and hence rotate, we may use Eq. \(3 / 50\) as long as the particle has no velocity relative to \(x-y-z\). (Hint: Use the first two terms of the binomial expansion for the approximation.)

A projectile is launched from \(B\) with a speed of \(2000 \mathrm{m} / \mathrm{s}\) at an angle \(\alpha\) of \(30^{\circ}\) with the horizontal as shown. Determine the maximum altitude \(h_{\max }\).

The slotted arm revolves in the horizontal plane about the fixed vertical axis through point \(O .\) The 3-lb slider \(C\) is drawn toward \(O\) at the constant rate of 2 in./sec by pulling the cord \(S\). At the instant for which \(r=9\) in., the arm has a counterclockwise angular velocity \(\omega=6\) rad/sec and is slowing down at the rate of 2 rad \(/\) sec \(^{2}\). For this instant, determine the tension \(T\) in the cord and the magnitude \(N\) of the force exerted on the slider by the sides of the smooth radial slot. Indicate which side, \(A\) or \(B\), of the slot contacts the slider.

The bungee jumper, an 80 -kg man, falls from the bridge at \(A\) with the bungee cord secured to his ankles. He falls \(20 \mathrm{m}\) before the 17 -m length of elastic bungee cord begins to stretch. The \(3 \mathrm{m}\) of rope above the elastic cord has no appreciable stretch. The man is observed to drop a total of \(44 \mathrm{m}\) before being projected upward. Neglect any energy loss and calculate \((a)\) the stiffness \(k\) of the bungee cord (increase in tension per meter of elongation) (b) the maximum velocity \(v_{\max }\) of the man during his fall, and \((c)\) his maximum acceleration \(a_{\max } .\) Treat the man as a particle located at the end of the bungee cord.
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