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Chapter Analysis
Intermediate33 pages • EnglishQuick Summary
The chapter on Systems of Particles and Rotational Motion explores the dynamics and kinematics of rigid bodies, emphasizing the concepts of center of mass, conservation of angular momentum, and the relations between torque, moment of inertia, and rotational motion. It draws analogies between translational and rotational motion to explain concepts such as angular velocity, acceleration, and equilibrium in mechanical systems. This chapter also includes practical examples of these concepts in everyday life and engineering applications.
Key Topics
- •Centre of mass
- •Linear momentum of a system of particles
- •Angular velocity and its relation with linear velocity
- •Torque and angular momentum
- •Equilibrium of a rigid body
- •Moment of inertia
- •Kinematics and dynamics of rotational motion about a fixed axis
Learning Objectives
- ✓Understand the concept of center of mass and its significance in motion analysis.
- ✓Relate angular momentum and torque to rotational dynamics.
- ✓Apply the principle of conservation of angular momentum to solve problems.
- ✓Differentiate between translational and rotational motion.
- ✓Use moment of inertia to quantify the rotational inertia of objects.
- ✓Analyze mechanical equilibrium using force and torque balance.
Questions in Chapter
Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?
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In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 10-10 m). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.
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A child sits stationary at one end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system?
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Show that the area of the triangle contained between the vectors a and b is one half of the magnitude of a × b.
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Show that a.(b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors a, b and c.
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Find the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components px, py and pz. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.
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Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the angular momentum vector of the two-particle system is the same whatever be the point about which the angular momentum is taken.
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Additional Practice Questions
What is the significance of the moment of inertia in rotational motion?
mediumAnswer: The moment of inertia in rotational motion is analogous to mass in translational motion. It quantifies how much torque is needed for a desired angular acceleration around a pivot point or axis. The larger the moment of inertia, the harder it is to change the rotational state of the object.
How does the conservation of angular momentum apply to a spinning skater?
mediumAnswer: A spinning skater can increase her rotation speed by pulling her arms close to her body. This decreases her moment of inertia and, by the conservation of angular momentum (L = Iω = constant if no external torque is applied), her angular velocity increases.
Explain how the concept of torque relates to rotational equilibrium.
mediumAnswer: Rotational equilibrium occurs when the net torque acting on a body is zero. This means the body is not experiencing any angular acceleration and will maintain its rotational state unless acted upon by an external torque.
Why is the center of mass important in understanding the motion of a system?
easyAnswer: The center of mass of a system is the point at which the total mass of the system can be assumed to be concentrated for analysis of translational motion, and about which the force exerted on the system produces no net torque.
Calculate the angular momentum of a particle moving in a circular path with constant speed.
easyAnswer: Angular momentum L of a particle moving in a circle of radius r with constant speed v is given by L = mvr, where m is the particle's mass.