In simple harmonic motion, the particle velocity lags behind the displacement by a phase angle of __________.
π/2
Understanding Phase Relationship in Simple Harmonic Motion (SHM)
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 the displacement. Key parameters in SHM include displacement, velocity, and acceleration, all of which vary sinusoidally with time but have different phase relationships.
Displacement in SHM
The displacement (\(x\)) of a particle in SHM from its mean position can be described by the equation:
\(x(t) = A \sin(\omega t + \phi)\)
Where:
- \(A\) is the amplitude (maximum displacement).
- \(\omega\) is the angular frequency.
- \(t\) is time.
- \(\phi\) is the initial phase or phase constant.
- The term \((\omega t + \phi)\) is the phase of the displacement at time \(t\).
Velocity in SHM
The velocity (\(v\)) of the particle in SHM is the rate of change of displacement with respect to time. We can find the velocity by differentiating the displacement equation:
\(v(t) = \frac{dx}{dt} = \frac{d}{dt} [A \sin(\omega t + \phi)]\)
\(v(t) = A \frac{d}{dt} [\sin(\omega t + \phi)]\)
Using the chain rule, \(\frac{d}{dt}[\sin(u)] = \cos(u) \frac{du}{dt}\), where \(u = \omega t + \phi\) and \(\frac{du}{dt} = \omega\):
\(v(t) = A \omega \cos(\omega t + \phi)\)
Phase Difference Calculation
To compare the phase of velocity with the phase of displacement, it's helpful to express the velocity equation using a sine function. We use the trigonometric identity: \(\cos(\theta) = \sin(\theta + \pi/2)\).
Applying this identity to the velocity equation:
\(v(t) = A \omega \sin((\omega t + \phi) + \pi/2)\)
Now we can compare the phase of the velocity equation with the phase of the displacement equation:
- Phase of displacement: \(\Phi_x = \omega t + \phi\)
- Phase of velocity: \(\Phi_v = \omega t + \phi + \pi/2\)
The phase difference (\(\Delta \Phi\)) between velocity and displacement is given by:
\(\Delta \Phi = \Phi_v - \Phi_x = (\omega t + \phi + \pi/2) - (\omega t + \phi)\)
\(\Delta \Phi = \pi/2\)
Understanding Leading and Lagging Phase Angles
A positive phase difference (\(\Delta \Phi > 0\)) means the first quantity (velocity in this case) leads the second quantity (displacement). A negative phase difference (\(\Delta \Phi < 0\)) means the first quantity lags behind the second quantity.
Our calculation shows \(\Delta \Phi = \pi/2\). This positive value indicates that the phase of velocity is \(\pi/2\) ahead of the phase of displacement. In other words, velocity leads displacement by a phase angle of \(\pi/2\).
If velocity leads displacement by \(\pi/2\), it is also true that displacement lags behind velocity by \(\pi/2\). The question asks by what phase angle velocity lags behind displacement. While the standard relationship is that velocity leads displacement by \(\pi/2\), the phase difference magnitude is indeed \(\pi/2\). Given the options and the likely intent of the question, it refers to the magnitude of the phase difference, where velocity is ahead of displacement.
Thus, the phase difference between particle velocity and displacement in simple harmonic motion is \(\pi/2\).
The correct phase angle is \(\pi/2\).
| Quantity | Equation | Phase |
|---|---|---|
| Displacement (\(x\)) | \(A \sin(\omega t + \phi)\) | \(\omega t + \phi\) |
| Velocity (\(v\)) | \(A\omega \sin(\omega t + \phi + \pi/2)\) | \(\omega t + \phi + \pi/2\) |
Revision Table: SHM Phase Relationships
| Relationship | Phase Difference | Leading/Lagging |
|---|---|---|
| Velocity relative to Displacement | \(\Phi_v - \Phi_x = +\pi/2\) | Velocity leads Displacement by \(\pi/2\) |
| Displacement relative to Velocity | \(\Phi_x - \Phi_v = -\pi/2\) | Displacement lags Velocity by \(\pi/2\) |
| Acceleration relative to Velocity | \(\Phi_a - \Phi_v = +\pi/2\) | Acceleration leads Velocity by \(\pi/2\) |
| Acceleration relative to Displacement | \(\Phi_a - \Phi_x = +\pi\) | Acceleration leads Displacement by \(\pi\) (or is in opposite phase) |
Additional Information on SHM Phase Angles
The phase differences between displacement, velocity, and acceleration in simple harmonic motion are crucial for understanding the system's dynamics. These constant phase shifts arise because velocity is the derivative of displacement, and acceleration is the derivative of velocity. Differentiation of a sine or cosine function shifts its phase by \(\pi/2\).
- When displacement is maximum (at amplitude), velocity is zero.
- When displacement is zero (at mean position), speed is maximum.
- Velocity is zero when acceleration is maximum (and in the opposite direction).
- Acceleration is maximum when displacement is maximum (and in the opposite direction), showing a \(\pi\) phase difference.
These relationships can be visualized using phasor diagrams or graphs of displacement, velocity, and acceleration versus time, which show how the peaks and zero points of each quantity are shifted relative to each other.
Important Questions from Simple Harmonic Motion
A pendulum clock is lifted to a height where the gravitational acceleration has a certain value g. Another pendulum clock of same length but of double the mass of the bob is lifted to another height where the gravitational acceleration is g/2. The time period of the second pendulum would be:
(in terms of period T of the first pendulum)- The displacement of a particle is given by $y(t) = K + P \sin^2(\omega t) + Q \sin(\omega t) \cos(\omega t)$. If this represents a simple harmonic motion, the amplitude of its oscillation is:
- Which one of the following equations of motion represents simple harmonic motion?
Assume $A$, $B$, $C$, $D$, $m$, $k$, and $\omega$ are all positive constants. - A particle undergoes simple harmonic motion. Determine the phase difference between its instantaneous velocity and instantaneous acceleration.
- A particle executes simple harmonic motion along a straight line. When its displacement from the mean position is $x$, its speed is $v$. If the displacement becomes $2x$, its speed reduces to $v/2$. What is the amplitude of the oscillation?
