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Questions and Answers
In Simple Harmonic Motion (SHM), what is the equation for displacement?
In Simple Harmonic Motion (SHM), what is the equation for displacement?
- $x(t) = -A\sin(\omega t)$
- $x(t) = -A\cos(\omega t)$
- $x(t) = A\sin(\omega t)$
- $x(t) = A\cos(\omega t)$ (correct)
What is the equation for velocity in Simple Harmonic Motion (SHM)?
What is the equation for velocity in Simple Harmonic Motion (SHM)?
- $v(t) = \omega A\cos(\omega t)$
- $v(t) = A\cos(\omega t)$
- $v(t) = -A\sin(\omega t)$
- $v(t) = -\omega A\sin(\omega t)$ (correct)
What is the equation for acceleration in Simple Harmonic Motion (SHM)?
What is the equation for acceleration in Simple Harmonic Motion (SHM)?
- $a(t) = -\omega^2 x(t)$ (correct)
- $a(t) = A\sin(\omega t)$
- $a(t) = -A\cos(\omega t)$
- $a(t) = \omega^2 x(t)$
What does the restoring force equation for Simple Harmonic Motion (SHM) represent?
What does the restoring force equation for Simple Harmonic Motion (SHM) represent?
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What does a larger spring constant ($k$) indicate in a Simple Harmonic Motion (SHM) system?
What does a larger spring constant ($k$) indicate in a Simple Harmonic Motion (SHM) system?
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Study Notes
Simple Harmonic Motion (SHM)
- SHM involves oscillatory motion, with linear velocity defined as the rate of change of displacement, expressed as dx/dt.
- Velocity as a function of time can be represented by the equation: ( v = -A\omega \sin(\omega t) ).
- The relationship between the square of the velocity and displacement is given by: ( v^2 = A^2\omega^2(1 - \cos^2(\omega t)) ), which simplifies to ( v^2 + \omega^2x^2 = A^2\omega^2 ).
- Maximum velocity, ( V_{\text{max}} ), occurs at the equilibrium position (x = 0) and is calculated as ( V_{\text{max}} = A\omega ).
Acceleration in SHM
- Linear acceleration is defined as the rate of change of velocity, represented as ( a = \frac{dv}{dt} ).
- The acceleration function can be derived from displacement as: ( a = -A\omega^2 \cos(\omega t) ).
- Maximum acceleration occurs when the displacement is at its maximum value (x = A), expressed as ( a_{\text{max}} = A\omega^2 ).
Force Relationships and SHM
- For a force to imply simple harmonic oscillation, it must be proportional to position x and act in the opposite direction, represented as ( F = -kx ).
- Options analyzed for SHM implication:
- (a) ( F = -5x ): Implies SHM (linear relationship).
- (b) ( F = -400x^2 ): Does not imply SHM (quadratic relationship).
- (c) ( F = 10x ): Implies SHM (linear relationship).
- (d) ( F = 3x^2 ): Does not imply SHM (quadratic relationship).
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