Test: Oscillations - Motion of Particle in SHM (7 Oct) - JEE MCQ

Time period (T): The time taken to complete one oscillation is called the time period.
The time period of a spring-mass system is given by:

where m is the mass, and k is the spring constant.
The parallel combination of spring: Figure a,b, and c shows the parallel combination of two springs for the spring-mass system.

The effective coefficient of spring in parallel combination is given by

• The Series combination of spring: The figure shows the series combination of two springs for the spring-mass system.

The effective coefficient of spring in series combination is given by:

CALCULATION:
We have to calculate the time period of spring-mass balance for which mass m is connected with two springs k₁ and k₂ in parallel:

For the parallel combination of springs k₁ and k₂

Hence the correct answer is option 1.

Concept -
For simple harmonic motion, the displacement equation must be of the form: x = Acos(ωt + φ)
Where A is amplitude, ω is angular frequency and φ is phase angle.
Explanation -
Given that the equation representing distance x traveled by the particle is: x = 2 cos at - sin (2a - 1)t
For simple harmonic motion, the displacement equation must be of the form: x = Acos(ωt + φ)
Where A is amplitude, ω is angular frequency and φ is phase angle.
Comparing the given equation to standard SHM equation:
There should only be one trigonometric term. Here there are two terms - cos and sin terms.
For them to reduce to the standard single term form, the coefficient of sin term must be 0.
Equating the coefficient of sin term to 0:  (2a - 1) = 0
⇒ a = 1/2
Therefore, the value of a that would make this motion simple harmonic is 1/2.

CONCEPT:

• Simple harmonic motion occurs when the restoring force or acceleration is directly proportional to the displacement from equilibrium or mean position.

F α -x
Where F is the force and x is the displacement from equilibrium.
The equation of displacement in SHM is given by:
x = A sin(ωt+ϕ)      .......(i)
where x is the displacement from the mean position at any time t, A is amplitude, t is time, ϕ is the initial phase and ω is the angular frequency.​
The equation of velocity in Simple harmonic motion is given by differentiating equation (i)
v = dx/dt = d (A sin(ωt + ϕ)) / dt
v = Aω cos(ωt + ϕ)
where v is the velocity at any time t, A is amplitude, t is time,  ϕ is the initial phase, and ω is the angular frequency.​
CALCULATION:
Given that equation of displacement is x = 10 sin(2πt + ϕ).
The equation of velocity can be obtained from the equation of displacement by differentiating the equation of velocity.
x = 10 sin(2πt + ϕ).
differentiate both sides with respect to 't'
v = dx/dt = d (10 sin(2πt + ϕ)) / dt
v = 20π cos(2πt + ϕ)
So the correct answer is option 3.

CONCEPT:

• Simple harmonic motion: The motion in which the restoring force is directly proportional to the displacement from equilibrium is simple harmonic motion.

⇒ F α -x
Where F = force and x = the displacement from mean position equilibrium.

• The equation of displacement in Simple Harmonic Motion is given by:

⇒ x = A sin(ωt + ϕ) .........(i)
where x is the distance from the mean position or equilibrium at any time t, A is amplitude (max displacement), ω is the angular frequency, and t is time.

• The equation of velocity in Simple Harmonic Motion is given by differentiating equation (i)

where v is the velocity at any time t, A is amplitude (max displacement), ω is the angular frequency, and t is time.

• The equation of acceleration in Simple Harmonic Motion is given by differentiating equation (ii)

where a is the acceleration at any time t, A is amplitude (max displacement), ω is the angular frequency, and t is time.
CALCULATION:

• Equation of velocity

⇒ v = Aω cos(ωt + ϕ)
⇒ v = Aω sin(ωt + ϕ + π/2)
Phase of displacement = ωt + ϕ + π/2

• Equation of acceleration is 

⇒ a = -Aω² sin(ωt + ϕ)
⇒ a = Aω² sin(ωt + ϕ + π)
Phase of acceleration = ωt + ϕ + π

• Difference between phase of acceleration and phase of velocity

⇒ Δϕ =  (ωt + ϕ + π) - (ωt + ϕ + π/2) = π/2 or 90° 

• So velocity lags acceleration by 90°. So the correct answer is option 2.

• Simple Harmonic Motion or SHM is a specific type of oscillation in which the restoring force is directly proportional to the displacement of the particle from the mean position.
• F ∝ y
• Acceleration, a = ω²x

• Where, T = time period, ω = angular frequency, y = displacement, F = force, a = acceleration, v = velocity,

Calculation:
Given, displacement, y = 3 cos 
Then, v = dy/dt
⇒ v = (-3) × (-2ω) sin 
⇒ v = 6ω sin 
Now 
⇒ a = -2ω × 6ω cos 
⇒ a = -12ω² × y
⇒ a ∝ -y.
So, the motion of the particle is Simple harmonic
Here, ω' = 2ω
Then 
Time period is 

• Simple harmonic motion occurs when the restoring force is directly proportional to the displacement from equilibrium.

F α -x
Where F = force and x = the displacement from equilibrium.
The equation of SHM is given by:
x = A sin(ωt + ϕ)
where x is the distance from the mean position at any time t, A is amplitude, t is time, ϕ is initial phase and ω is the angular frequency.

• The amplitude of SHM (A): maximum displacement from the mean position.
• frequency (f): no. of oscillations in one second.
• Time period (T): time taken to complete one oscillation.

The relation between time period (T) and frequency (f) is given by:

Angular frequency (ω) of SHM is given by:

where T is the time period.
EXPLANATION:
If the simple harmonic motion is represented by x = A cos(ωt + φ), then 'ω' is the angular frequency.
So the correct answer is option 3.

Simple harmonic motion occurs when the restoring force is directly proportional to the displacement from equilibrium.
F α -x
Where F = force and x = the displacement from equilibrium.
The equation of displacement in SHM is given by:
x = A Cos(ωt+ϕ) .........(i)
where x is the distance from the mean position at any time t, A is amplitude, t is time, and ω is the angular frequency.​
The equation of velocity in SHM is given by differentiating equation (i)
v = dx/dt = d (A Cos(ωt+ϕ)) / dt
v = - Aω Sin(ωt+ϕ)
where v is the velocity at any time t, A is amplitude, t is time, and ω is angular frequency.​
In the same way, the equation of displacement can be obtained from the equation of velocity by integrating the equation of velocity.
CALCULATION:
v = - Aω Sin(ωt+ϕ)
The acceleration is given by:
a = dv/dt = d (- Aω Sin(ωt+ϕ))/dt = –ω²A cos(ωt + φ).
So option 1 is correct.

• Simple harmonic motion occurs when the restoring force is directly proportional to the displacement from equilibrium.

F α -x
Where F = force and x = the displacement from equilibrium.

• For a simple harmonic motion equation of acceleration

a = -ω²x
where a is the acceleration ω is the angular frequency and x is the displacement.

• Time period (T): time taken to complete one oscillation.

The relation between time period (T) and frequency (f) is given by:
f = 1/T

• Angular frequency (ω) of SHM is given by:

where T is the time period.
CALCULATION:
Given that Equation of motion is a = -bx, where a is the acceleration, x is the displacement from the mean position, and b any constant.
Compare it with the equation of acceleration a = -ω²x
ω² = b
ω = √b

So the correct answer is option 3.

• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
  • Example: Motion of an undamped pendulum, undamped spring-mass system.

Force (F) = - k x
Acceleration (a) = - (k/m) x
Where a is acceleration, x is the displacement of the system from its equilibrium position, m is the mass of the system and k is a constant associated with the system.
EXPLANATION:

• In simple harmonic motion, the acceleration is proportional to the distance from the point of reference and directed towards it. So option 3 is correct.

CONCEPT:

• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

Force (F) = - k x
Where k = restoring force, x = distance from the equilibrium position, F = force it experiences towards mean position

• Example: Motion of an undamped pendulum, undamped spring-mass system.

​EXPLANATION:

• Simple harmonic motion is represented by

⇒ x = A cos(ωt + φ)
Where A = amplitude, ω = Angular frequency, x = Displacement and φ = Phase constant