Questions & explanations
1. Which of the following motions is an example of simple harmonic motion (SHM)?
- A) The motion of a planet around the Sun
- B) The motion of a simple pendulum for small angular displacements
- C) The motion of a freely falling body under gravity
- D) The motion of a ball bouncing elastically on a hard floor
Answer: B) The motion of a simple pendulum for small angular displacements
Simple harmonic motion occurs when the restoring force is proportional to the displacement and acts opposite to it. For a simple pendulum with small amplitude, the restoring torque is approximately proportional to the angular displacement, making it SHM. Planetary motion is periodic but not oscillatory; free fall and bouncing ball do not have a restoring force proportional to displacement.
2. Which statement is correct about the total mechanical energy of a particle executing simple harmonic motion?
- A) It varies sinusoidally with time
- B) It remains constant throughout the motion
- C) It is maximum at the mean position and zero at extremes
- D) It is proportional to the square of the frequency
Answer: B) It remains constant throughout the motion
In ideal SHM (no damping), the total mechanical energy is conserved; it only transforms between kinetic and potential forms. Option A is false because energy is constant; C is false because total energy is same at all positions; D is true but not the best general statement—constant is the key property asked. The question asks which statement is correct; B is directly correct.
3. A particle moving uniformly on a circle of radius R with constant angular speed ω has its projection on a diameter performing simple harmonic motion. The acceleration (a) of the projection is related to its displacement (x) from the mean position as:
- a = ω²x
- a = -ω²x
- a = ω²R
- a = -ω²R
Answer: a = -ω²x
The projection of uniform circular motion onto a diameter executes simple harmonic motion. In SHM, acceleration is directly proportional to the displacement from the mean position and directed opposite to it. The angular frequency of the SHM equals the uniform angular speed ω of the circular motion, giving the relation a = -ω²x.
4. For a mass m attached to a spring of force constant k, the time period of oscillation in SHM is:
- T = 2π√(k/m)
- T = 2π√(m/k)
- T = 2π√(m/k) only if amplitude is small
- T = (1/2π)√(m/k)
Answer: T = 2π√(m/k)
The time period of a mass-spring system is derived from the equation of motion and is given by T = 2π√(m/k). Option A is the reciprocal, option C incorrectly adds a condition (small amplitude is not needed for a mass-spring system as it is SHM for any amplitude), and option D is missing the factor of 2π in the numerator.
5. In forced oscillations, resonance occurs when the frequency of the driving force is:
- Less than the natural frequency of the system
- Equal to the natural frequency of the system
- Greater than the natural frequency of the system
- Independent of the natural frequency
Answer: Equal to the natural frequency of the system
Resonance is the phenomenon where the amplitude of forced oscillations is maximum. This happens when the driving frequency matches the natural frequency of the system. Options A and C give conditions where amplitude is less than maximum, and option D is incorrect because resonance depends on the natural frequency.
6. In which type of oscillation does the amplitude continuously decrease with time due to dissipative forces like friction?
- Free oscillations
- Damped oscillations
- Forced oscillations
- Resonance
Answer: Damped oscillations
Damped oscillations occur when dissipative forces (e.g., friction, air resistance) cause the amplitude to decrease over time. Free oscillations ideally have constant amplitude, forced oscillations are driven by an external periodic force, and resonance is a condition of maximum amplitude in forced oscillations.
7. Which of the following correctly describes the acceleration a of a particle in simple harmonic motion as a function of its displacement x from the mean position?
- A) a ∝ x, in the direction of x
- B) a ∝ x, opposite to x
- C) a ∝ -x, in the direction of x
- D) a ∝ -x, opposite to x
Answer: D) a ∝ -x, opposite to x
In SHM, acceleration is proportional to displacement and directed opposite to it, i.e., a = -ω²x. The negative sign indicates that acceleration always points toward the mean position, opposite to the displacement. Options A and B lack the negative sign; option C incorrectly says 'in the direction of x'.
8. A particle of mass m executes SHM with amplitude A and angular frequency ω. Its total mechanical energy is:
- A) (1/2) m ω² A²
- B) m ω² A²
- C) (1/2) m ω A²
- D) (1/2) m ω² A
Answer: A) (1/2) m ω² A²
Total mechanical energy in SHM is the sum of kinetic and potential energies and equals maximum kinetic energy (or maximum potential energy). At maximum displacement, KE=0 and PE = (1/2)kA²; using k = mω² gives total energy = (1/2)mω²A². The other options have incorrect factors or dimensions.
9. In simple harmonic motion, the phase difference between displacement and velocity is:
- A) 0
- B) π/2
- C) π
- D) 3π/2
Answer: B) π/2
If displacement is x = A cos(ωt), then velocity v = -Aω sin(ωt) = Aω cos(ωt + π/2). Thus velocity leads displacement by a phase of π/2 (or displacement lags velocity by π/2). The other phase differences are incorrect; for example, π corresponds to acceleration being opposite to displacement.
10. Two springs of spring constants k₁ and k₂ are connected in series. The effective spring constant of the combination is:
- k₁ + k₂
- (k₁ k₂)/(k₁ + k₂)
- (k₁ + k₂)/(k₁ k₂)
- √(k₁ k₂)
Answer: (k₁ k₂)/(k₁ + k₂)
For springs in series, the reciprocal of the effective spring constant is the sum of reciprocals: 1/k_eff = 1/k₁ + 1/k₂. Hence k_eff = (k₁ k₂)/(k₁ + k₂). Option A is for parallel, option C is the reciprocal, and option D is the geometric mean (not correct for series).
11. In SHM, the velocity v of the particle when its displacement from mean position is x is given by (A = amplitude, ω = angular frequency):
- A) v = ω√(A² - x²)
- B) v = ωx
- C) v = ω√(A² + x²)
- D) v = ω²√(A² - x²)
Answer: A) v = ω√(A² - x²)
From energy conservation or differentiation, the velocity in SHM is v = ω√(A² - x²). It decreases as x increases and is maximum at x = 0. Option B corresponds to uniform motion; C gives a value larger than possible; D has wrong dimensions.
12. A mass m attached to a spring of force constant k oscillates with amplitude A. The total mechanical energy of the system is:
- (1/2) k A²
- (1/2) m A² ω² (where ω is the angular frequency)
- Both (1/2) k A² and (1/2) m A² ω² are correct
- (1/2) k A
Answer: Both (1/2) k A² and (1/2) m A² ω² are correct
In SHM, total mechanical energy is constant and equals (1/2) k A². Since ω² = k/m, (1/2) m A² ω² = (1/2) m A² (k/m) = (1/2) k A², so both expressions are equivalent and correct. Option D is dimensionally incorrect (missing a factor of A).