38 Questions
What is the physical significance of the term A in the equation of simple harmonic motion?
It represents the maximum displacement of the particle
What is the physical significance of the term φ in the equation of simple harmonic motion?
It is the phase constant of the motion
What is the relation between ω and T in simple harmonic motion?
ω is inversely proportional to T
What is the physical significance of the term (ωt + φ) in the equation of simple harmonic motion?
It is the phase of the motion at time t
What can be said about two simple harmonic motions with the same A and ω but different φ?
They have different phase angles
What is the unit of the angular frequency ω in simple harmonic motion?
radians per second
What is the significance of taking φ = 0 in the equation of simple harmonic motion?
It simplifies the equation of motion
What is the physical significance of the graph shown in Fig. 13.7(b)?
It shows two simple harmonic motions with the same amplitude and angular frequency but different phase angles
What is the angular speed of the particle P?
ω rad/s
What is the initial angle made by the position vector OP with the positive direction of the x-axis?
φ
What is the angle made by the position vector OP with the positive direction of the x-axis at time t?
ωt + φ
What is the projection of the position vector OP on the x-axis?
OP′
What is the expression for sin(ωt) - cos(ωt) in terms of ωt?
√2 sin(ωt - π/4)
What is the sense of rotation of the particle P?
Anticlockwise
What is the radius of the circle on which the particle P is moving?
A
What is the angle covered by the particle P in time t?
ωt
What is the nature of the velocity of a particle executing SHM?
Periodic function of time
At what position is the kinetic energy of a particle in SHM zero?
At the extreme positions of displacement
What is the relationship between kinetic and potential energies of a particle in SHM?
They vary between zero and their maximum values
What is the direction of the displacement of the mass in Fig. 13.15?
To the right side of the equilibrium position
What is the definition of kinetic energy (K) of a particle in SHM?
It is defined as the product of mass and velocity
What can be said about the kinetic energy and potential energy of a particle in SHM?
They vary between zero and their maximum values
What is the angle made by OP with the x-axis at time t = 0?
45°
What is the angle covered by the particle P in the anticlockwise sense after time t?
2πt
What is the expression for the displacement of the particle P′ on the x-axis?
x(t) = A cos(ωt + φ)
What is the period of the SHM given by the equation x(t) = A cos(2πt/4 + π/4)?
4 s
What is the phase difference between the projections of the motion of P on the x-axis and the y-axis?
π/2
What is the expression for the displacement of the particle P′ on the y-axis?
y(t) = A sin(ωt + φ)
What is the amplitude of the SHM given by the equation x(t) = A cos(2πt/4 + π/4)?
A
What is the force acting on a particle in linear simple harmonic motion?
A force different from centripetal force
What is the total mechanical energy of a simple harmonic motion system if no friction is present?
Constant and remains the same
What type of force is responsible for simple harmonic motion?
Hooke's law restoring force
What is the period of oscillation of a simple pendulum swinging through small angles?
T = 2π√(L/g)
What is the characteristic of a linear oscillator?
Governed by Hooke's law restoring force
What is the condition for a periodic motion to be simple harmonic?
The force must be proportional to the displacement
What is the relation between the kinetic energy and potential energy of a simple harmonic motion system?
K + U = E
What is the condition for circular motion to arise due to a simple harmonic force?
The phases of motion in two perpendicular directions must differ by π/2
What is the period of a simple harmonic motion system?
T = 2π√(m/k)
Study Notes
Simple Harmonic Motion (SHM)
- SHM is a type of periodic motion where the particle moves back and forth about its equilibrium position.
- The amplitude (A) of SHM is the magnitude of the maximum displacement of the particle.
- The phase (ωt + φ) of SHM is the quantity that determines the initial position of the particle at t = 0.
- The phase constant (φ) is the value of the phase at t = 0.
Angular Frequency (ω)
- The angular frequency (ω) is related to the period of motion (T) by the equation ω = 2π / T.
- The angular frequency (ω) is also related to the period of revolution of a particle moving uniformly on a circle.
Simple Harmonic Motion Equation
- The equation of SHM is x(t) = A cos (ωt + φ).
- The equation of SHM can also be written in terms of sine function as x(t) = A sin (ωt + φ + π/2).
Projection of Circular Motion on a Diameter
- When a particle moves uniformly on a circle, its projection on a diameter executes SHM.
- The projection of the motion on the x-axis is given by x(t) = A cos (ωt + φ).
- The projection of the motion on the y-axis is given by y(t) = A sin (ωt + φ), which is also an SHM with the same amplitude but differing by a phase of π/2.
Energy in Simple Harmonic Motion
- The kinetic energy (K) of a particle in SHM varies between zero and its maximum value.
- The potential energy (U) of a particle in SHM varies between zero and its maximum value.
- The mechanical energy (E) of a particle in SHM remains constant, E = K + U.
Period and Frequency of SHM
- The period (T) of SHM is the time taken by the particle to complete one oscillation.
- The frequency (f) of SHM is the number of oscillations per second, f = 1 / T.
- The angular frequency (ω) is related to the period by ω = 2π / T.
Points to Ponder
- The period (T) is the least time after which the motion repeats itself.
- Not every periodic motion is SHM; only that periodic motion governed by the force law F = – k x is SHM.
- Circular motion can arise due to an inverse-square law force or due to SHM in two dimensions.
Understand the concepts of Simple Harmonic Motion, including amplitude, phase, and phase constant. Learn how to determine the amplitude and phase constant from the displacement at t = 0.
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