91Ó°ÊÓ

In Simple Harmonic Motion (SHM), understanding angular frequency is vital. Angular frequency, denoted by \( \omega \), relates to how quickly an object oscillates back and forth. It's directly connected to the standard frequency, \( f \), and is calculated using the formula: \[ \omega = 2\pi f \] This equation shows that angular frequency is a measure of angular displacement per unit time, often expressed in radians per second. For the tuning fork with a frequency of 1000 Hz, the angular frequency becomes: \[ \omega = 2\pi \times 1000 = 2000\pi \text{ rad/s} \] It's important to understand that higher frequencies mean more oscillations per second, leading to a larger angular frequency.

The maximum acceleration in SHM occurs when the object is at its extreme positions. This means when the object moves the furthest from its equilibrium. It's calculated using the formula: \[ a_{max} = \omega^2 A \] where \( A \) is the amplitude, the maximum displacement from the equilibrium position.In this case: - \( \omega = 2000\pi \text{ rad/s} \) - \( A = 0.40 \text{ mm} = 0.0004 \text{ m} \) Thus, the maximum acceleration is: \[ a_{max} = (2000\pi)^2 \times 0.0004 = 5.02 \times 10^3 \text{ m/s}^2 \] This significant value illustrates how quickly the acceleration reaches its peak during the motion.

In the context of SHM, maximum velocity is reached as the object passes through its equilibrium position with the highest speed. We determine maximum velocity using the formula: \[ v_{max} = \omega A \] It's crucial to note that this happens exactly when all the potential energy is converted to kinetic energy.Given: - \( \omega = 2000\pi \text{ rad/s} \) - \( A = 0.0004 \text{ m} \) This sets up the calculation for maximum velocity as follows: \[ v_{max} = 2000\pi \times 0.0004 = 2.51 \text{ m/s} \] This value represents the fastest speed of the tuning fork's prong during its oscillation.

Velocity in SHM is not constant; it varies depending on the displacement from the equilibrium point. To find velocity at a particular displacement, we use: \[ v = \pm \omega \sqrt{A^2 - x^2} \] where \( x \) is the current displacement. This gives the speed at which the object moves at a specific point.For our tuning fork: - \( A = 0.0004 \text{ m} \) - \( x = 0.0002 \text{ m} \) - \( \omega = 2000\pi \text{ rad/s} \) Inserting these values, we get: \[ v = \pm 2000\pi \sqrt{(0.0004)^2 - (0.0002)^2} = \pm 2.17 \text{ m/s} \] This indicates the speed of the tuning fork's prong at \( 0.20 \text{ mm} \) displacement can be either positive or negative, reflecting its direction.

Acceleration in SHM is always directed towards the equilibrium position and varies depending on the displacement. For a given displacement \( x \), we find acceleration using: \[ a = -\omega^2 x \] This reveals how quickly the velocity of the object changes at that point, always acting to restore the object back to equilibrium.In our situation: - \( \omega = 2000\pi \text{ rad/s} \) - \( x = 0.0002 \text{ m} \) The acceleration is then calculated as: \[ a = -(2000\pi)^2 \times 0.0002 = -2.51 \times 10^3 \text{ m/s}^2 \] The negative sign indicates that the acceleration is directed towards the equilibrium, opposing the displacement.