SHM Amplitude Calculation Using Acceleration & Velocity
This problem involves understanding the relationship between key parameters in Simple Harmonic Motion (SHM). We are given the angular velocity ($\omega$) and the maximum acceleration ($a_{max}$) of a body undergoing SHM, and we need to find its amplitude (A).
Understanding the Concepts
Simple Harmonic Motion (SHM) is a special type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. In SHM, the acceleration of the body is proportional to its displacement from the equilibrium position and is directed towards the equilibrium position.
The relationship between maximum acceleration ($a_{max}$), amplitude (A), and angular velocity ($\omega$) in SHM is given by the formula:
$a_{max} = A \omega^2$
Solving for Amplitude
To find the amplitude (A), we can rearrange the formula:
$A = \frac{a_{max}}{\omega^2}$
Step-by-Step Calculation
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Identify Given Values:
The problem provides:
- Angular velocity, $\omega = 2 \text{ rad/s}$
- Maximum acceleration, $a_{max} = 0.8 \text{ m/s}^2$
- Apply the Formula: We use the rearranged formula $A = \frac{a_{max}}{\omega^2}$.
- Substitute Values: Plug the given values into the formula: $A = \frac{0.8 \text{ m/s}^2}{(2 \text{ rad/s})^2}$
- Calculate the Result: First, calculate the square of the angular velocity: $(2 \text{ rad/s})^2 = 4 \text{ rad}^2/\text{s}^2$ Now, perform the division: $A = \frac{0.8 \text{ m/s}^2}{4 \text{ rad}^2/\text{s}^2}$ $A = 0.2 \text{ m}$
- Final Answer: The amplitude of the oscillations of the body is $0.2 \text{ m}$.
This calculated value matches one of the provided options.
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:
- 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. 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?
