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Chapter Analysis
Intermediate15 pages • EnglishQuick Summary
The chapter on Oscillations covers the concept of periodic and oscillatory motion, with a focus on simple harmonic motion (SHM). It explains SHM as the projection of uniform circular motion and discusses detailed mathematical representations of displacement, velocity, and acceleration in SHM. The chapter also covers the energy conservation in harmonic oscillators and the relationship between SHM and forces like Hooke's Law. Essential concepts include the period, frequency, and phase of oscillations, with examples such as the simple pendulum.
Key Topics
- •Periodic and oscillatory motion
- •Simple harmonic motion (SHM)
- •Dynamics of SHM
- •Energy in SHM
- •Simple pendulum
- •Phase and frequency relationships
- •Restoring forces
- •Connection between SHM and circular motion
Learning Objectives
- ✓Define and identify periodic and oscillatory motions.
- ✓Understand the mathematical representation of simple harmonic motion.
- ✓Analyze the energy transformations in simple harmonic oscillators.
- ✓Relate SHM to uniform circular motion and derive equations.
- ✓Calculate the period of oscillations for various systems.
- ✓Explain the effects of damping and external forces on oscillatory systems.
Questions in Chapter
Which of the following examples represent periodic motion? (a) A swimmer completing one (return) trip from one bank of a river to the other and back. (b) A freely suspended bar magnet displaced from its N-S direction and released. (c) A hydrogen molecule rotating about its centre of mass. (d) An arrow released from a bow.
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Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion? (a) the rotation of earth about its axis. (b) motion of an oscillating mercury column in a U-tube. (c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point. (d) general vibrations of a polyatomic molecule about its equilibrium position.
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Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (ω is any positive constant): (a) sin ωt – cos ωt, (b) sin3 ωt, (c) 3 cos (π/4 – 2ωt), (d) cos ωt + cos 3ωt + cos 5ωt, (e) exp (–ω2t2), (f) 1 + ωt + ω2t2.
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A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is (a) at the end A, (b) at the end B, (c) at the mid-point of AB going towards A, (d) at 2 cm away from B going towards A, (e) at 3 cm away from A going towards B, and (f) at 4 cm away from B going towards A.
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Which of the following relationships between the acceleration a and the displacement x of a particle involve simple harmonic motion? (a) a = 0.7x, (b) a = –200x2, (c) a = –10x, (d) a = 100x3.
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Additional Practice Questions
What is the definition of simple harmonic motion?
easyAnswer: Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
How is the period of a simple pendulum related to its length and the acceleration due to gravity?
mediumAnswer: The period T of a simple pendulum is given by T = 2π√(L/g), where L is the length of the pendulum, and g is the acceleration due to gravity.
Explain how the concept of phase applies to simple harmonic motion.
mediumAnswer: In simple harmonic motion, the phase is the angular quantity (ωt + φ) where ω is the angular frequency, t is the time, and φ is the phase constant. It determines the position and direction of motion of the oscillating particle.
How can simple harmonic motion be visualized using uniform circular motion?
mediumAnswer: Simple harmonic motion can be visualized as the projection of uniform circular motion on a diameter. As a particle moves in uniform circular motion, its projection on a diameter executes SHM.
Describe the energy changes in a simple harmonic oscillator.
mediumAnswer: In a simple harmonic oscillator, energy oscillates between kinetic and potential forms. At the equilibrium position, kinetic energy is maximum, while at the extreme positions, potential energy is maximum. Total energy remains constant.
What role does damping play in oscillatory systems?
hardAnswer: Damping refers to the effect of energy loss in an oscillatory system, typically due to friction or resistance, which results in the gradual reduction of amplitude over time.