Assume $A$, $B$, $C$, $D$, $m$, $k$, and $\omega$ are all positive constants.
Understanding Simple Harmonic Motion (SHM) Equations
Simple Harmonic Motion (SHM) is a fundamental concept in physics describing a special type of periodic oscillation. The defining characteristic of SHM is that the acceleration of the oscillating object is always directed towards a fixed equilibrium position and is directly proportional to the object's distance (displacement) from that equilibrium position. The mathematical condition for SHM is often expressed through a second-order linear differential equation.
The standard differential equation for SHM is:
$ \frac{d^2x}{dt^2} = -\omega^2 x $or equivalently,
$ \frac{d^2x}{dt^2} + \omega^2 x = 0 $Here, '$x$' represents the displacement from the equilibrium position, '$t$' is time, and '$\omega$' is the angular frequency of the motion (a positive constant). The negative sign is crucial; it signifies that the acceleration is always in the opposite direction to the displacement, acting as a restoring force pulling the object back towards equilibrium.
Analyzing the Motion Equations
Let's examine each provided equation to determine which one fits the definition of SHM. Remember that '$A$, $B$, $C$, $D$, $m$, $k$, and $\omega$' are given as positive constants.
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Option 1: $m \frac{d^2x}{dt^2} = -k(x + A)$
We can rewrite this equation by dividing by '$m$':
$ \frac{d^2x}{dt^2} = -\frac{k}{m}(x + A) $To see if this represents SHM, let's introduce a new variable, '$y$', representing the displacement from a potential equilibrium point. Let $y = x + A$. If we differentiate '$y$' twice with respect to time, we get $\frac{d^2y}{dt^2} = \frac{d^2x}{dt^2}$. Substituting this into the equation gives:
$ \frac{d^2y}{dt^2} = -\frac{k}{m}y $This equation is in the standard form of SHM, $\frac{d^2y}{dt^2} = -\omega^2 y$, where $\omega^2 = \frac{k}{m}$. The motion is simple harmonic about the equilibrium position $y=0$, which corresponds to $x = -A$. Therefore, this equation represents SHM.
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Option 2: $\frac{d^2x}{dt^2} + B \frac{dx}{dt} + \omega^2 x = 0$
This equation can be written as:
$ \frac{d^2x}{dt^2} = -B \frac{dx}{dt} - \omega^2 x $The term $B \frac{dx}{dt}$ represents damping (friction or resistance), which opposes the velocity $\frac{dx}{dt}$. Simple Harmonic Motion does not include damping. Since '$B$' is a positive constant, this equation describes damped harmonic motion, not SHM.
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Option 3: $acceleration $= C x^3 - D x$
This equation states:
$ \frac{d^2x}{dt^2} = C x^3 - D x $For SHM, the acceleration must be linearly proportional to the displacement '$x$' and directed towards the equilibrium position (i.e., $\frac{d^2x}{dt^2} \propto -x$). This equation includes a term $C x^3$. Since '$C$' is a positive constant, if '$C$' is non-zero, the acceleration is not simply proportional to '$x$'. This represents anharmonic motion (non-linear oscillation).
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Option 4: $m \frac{d^2x}{dt^2} = k(x - A)$
Rewriting the equation:
$ \frac{d^2x}{dt^2} = \frac{k}{m}(x - A) $Let $z = x - A$. Then $\frac{d^2z}{dt^2} = \frac{d^2x}{dt^2}$. The equation becomes:
$ \frac{d^2z}{dt^2} = \frac{k}{m}z $Since $\frac{k}{m}$ is positive, this equation indicates that the acceleration ($ \frac{d^2z}{dt^2} $) is in the same direction as the displacement ($z$). This leads to exponential growth or decay, not oscillatory motion. This system would move away from the equilibrium point $x=A$, rather than oscillating around it.
- Option 5: This option is empty and therefore cannot represent any physical motion.
Identifying the Correct SHM Equation
Based on the analysis, only Option 1 satisfies the condition for Simple Harmonic Motion. It can be transformed into the standard SHM differential equation $\frac{d^2y}{dt^2} = -\omega^2 y$ by a suitable change of variable, indicating oscillation around an equilibrium position ($x = -A$).
Key characteristics of SHM equation checked:
- The second derivative of displacement (acceleration) must be proportional to the negative of the displacement from equilibrium.
- The equation must be a second-order linear differential equation with constant coefficients.
- Terms involving higher powers of displacement ($x^3$) or velocity ($ \frac{dx}{dt} $) indicate deviations from SHM (anharmonic or damped motion).
Therefore, the equation $m \frac{d^2x}{dt^2} = -k(x + A)$ correctly represents simple harmonic motion.
Important Questions from Simple Harmonic Motion
- 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:
In simple harmonic motion, the particle velocity lags behind the displacement by a phase angle of __________.
- 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?
A particle executes simple harmonic motion with amplitude $A$ and time period $T$. If the particle starts its motion from one of its extreme positions, what is the total distance covered by the particle in the first $\frac{T}{6}$ of its motion?
