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Chapter 4: Problem 16

The two small spheres, each of mass \(m\), and their connecting rod of negligible mass are rotating about their mass center \(G\) with an angular velocity \(\omega\). At the same instant the mass center has a velocity \(v\) in the \(x\) -direction. Determine the angular momentum \(\mathbf{H}_{O}\) of the assembly at the instant when \(G\) has coordinates \(x\) and \(y\).

Short Answer

Expert verified
The angular momentum \( \mathbf{H}_O \) is the sum of translational and rotational components: \( \mathbf{H}_O = 2m ( \vec{r}_G \times \vec{v} ) + 2mr^2 \omega \).

Step by step solution

01

Identify the System

We have two small spheres, each with mass \( m \), connected by a rod with negligible mass. They are rotating around their mass center \( G \) with an angular velocity \( \omega \). The system is moving in the \( x \)-direction with a velocity \( v \).
02

Use Angular Momentum Definition

The angular momentum \( \mathbf{H}_O \) about a point \( O \) is the sum of the angular momentum due to the velocity of mass center \( G \) and the angular momentum due to the rotation of the spheres around \( G \).
03

Calculate Angular Momentum due to Translation

We calculate the translational component of the angular momentum. Since \( G \) has a center velocity \( v \) and the entire mass \( 2m \), \[ \mathbf{H}_{O, \text{trans}} = 2m \cdot ( \vec{r}_G \times \vec{v} ) \]where \( \vec{r}_G \) is the position vector from \( O \) to \( G \).
04

Calculate Angular Momentum due to Rotation

The rotational component of angular momentum is given by:\[ \mathbf{H}_{O, \text{rot}} = I_G \omega \]for the rotation about the mass center, where \( I_G \) is the moment of inertia of the spheres about the center \( G \). Based on the configuration, each sphere is at a distance \( r \), so:\[ I_G = 2m \cdot r^2 \]
05

Sum Translational and Rotational Components

Combine the components for total angular momentum relative to \( O \):\[ \mathbf{H}_O = \mathbf{H}_{O, \text{trans}} + \mathbf{H}_{O, \text{rot}} \]
06

Express and Combine Results

Plug in computed values:\[ \mathbf{H}_O = 2m ( \vec{r}_G \times \vec{v} ) + 2m \cdot r^2 \omega \]This equation accounts for both the translational and rotational movement contributions to the angular momentum.

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Key Concepts

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

Rotational Motion
Rotational motion refers to the movement of an object around a central point or axis. In this exercise, two spheres are connected by a rod and rotate about their mass center, denoted as \( G \). This is a classic example of rotational motion. The spheres have an angular velocity \( \omega \), which describes how quickly they rotate around \( G \). In rotational motion:
  • The angular position, speed, and acceleration are key factors.
  • The angular velocity \( \omega \) indicates the object's rotation rate and is measured in radians per second.
  • The direction of rotation also affects other physical properties, like angular momentum.
This concept is crucial for understanding how the rotational component contributes to the system's overall angular momentum.
Moment of Inertia
The moment of inertia, denoted \( I \), is a fundamental concept in rotational dynamics. It measures an object's resistance to changes in its rotational speed. Think of it like mass in linear motion. Here, each sphere contributes to the moment of inertia about the center \( G \). The formula for moment of inertia for point masses in this setup is:
\[ I_G = 2m \cdot r^2 \]where \( r \) is the distance from the axis (center) to the point mass. Key points include:
  • It depends not only on the mass but also on the distribution of mass relative to the axis.
  • Larger distances \( r \) result in greater moments of inertia.
  • It's crucial for calculating rotational components of angular momentum.
Understanding \( I \) helps us see how the spheres' rotation affects the entire system's dynamics.
Translational Motion
Translational motion involves the movement of an object from one place to another. In this exercise, the entire system (spheres and rod) has a velocity \( v \) in the \( x \)-direction. It illustrates how objects travel along a specific path, independent of rotation. Key features of translational motion include:
  • Velocity \( v \) determines the speed and direction of the system's mass center \( G \).
  • It contributes to the angular momentum through a different process than rotation does.
  • The expression for its contribution is \( \mathbf{H}_{O, \text{trans}} = 2m \cdot ( \vec{r}_G \times \vec{v} ) \).
While separate from rotation, translational motion provides a comprehensive view of the system's overall motion and helps calculate the total angular momentum efficiently.

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Most popular questions from this chapter

A model rocket weighs 1.5 lb just before its vertical launch. Its experimental solid-fuel motor carries 0.1 lb of fuel, has an escape velocity of 3000 ft/sec, and burns the fuel for 0.9 sec. Determine the acceleration of the rocket at launch and its burnout velocity. Neglect aerodynamic drag and state any other assumptions.

A tank truck for washing down streets has a total weight of 20,000 lb when its tank is full. With the spray turned on, 80 lb of water per second issue from the nozzle with a velocity of 60 ft/sec relative to the truck at the \(30^{\circ}\) angle shown. If the truck is to accelerate at the rate of \(2 \mathrm{ft} / \mathrm{sec}^{2}\) when starting on a level road, determine the required tractive force \(P\) between the tires and the road when \((a)\) the spray is turned on and ( \(b\) ) the spray is turned off.

The cars of a roller-coaster ride have a speed of \(30 \mathrm{km} / \mathrm{h}\) as they pass over the top of the circular track. Neglect any friction and calculate their speed \(v\) when they reach the horizontal bottom position. At the top position, the radius of the circular path of their mass centers is \(18 \mathrm{m},\) and all six cars have the same mass.

The carriage of mass \(2 m\) is free to roll along the horizontal rails and carries the two spheres, each of mass \(m,\) mounted on rods of length \(l\) and negligible mass. The shaft to which the rods are secured is mounted in the carriage and is free to rotate. If the system is released from rest with the rods in the vertical position where \(\theta=0,\) determine the velocity \(v_{x}\) of the carriage and the angular velocity \(\dot{\theta}\) of the rods for the instant when \(\theta=180^{\circ} .\) Treat the carriage and the spheres as particles and neglect any friction.

A small rocket-propelled vehicle weighs \(125 \mathrm{lb}\), in cluding 20 lb of fuel. Fuel is burned at the constant rate of 2 lb/sec with an exhaust velocity relative to the nozzle of \(400 \mathrm{ft} / \mathrm{sec}\). Upon ignition the vehicle is released from rest on the \(10^{\circ}\) incline. Calculate the maximum velocity \(v\) reached by the vehicle. Neglect all friction.
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