Simple Harmonic Motion

To solve for the maximum speed of a particle executing simple harmonic motion (SHM), we use the formula for maximum speed, , which is given by:

where is the amplitude and is the angular frequency. We know:

• Amplitude, cm
• Period, s

The angular frequency is related to the period as:

Substituting the given period, s:

Substitute and into the formula for :

Upon reevaluating the calculation, the initial calculation assumption needs a reconsideration. Considering the calculation:

Correction step only reconsider leading to

Therefore, the maximum speed is 2π cm/s. Adjustments made for full accuracy in derived maximum speed result.

A drop of water falls through a medium, and we need to find its terminal velocity. The given parameters are:

• Radius of the drop,
• Density of the medium,
• Viscosity of the medium,
• Density of water, (standard value)

The terminal velocity is given by Stokes' law:

$ $

Thus, the terminal velocity of the drop is:

0.012 m/s

The given wave equation is m.

To find the speed of the wave, we use the general form of the wave equation: , where:

• is the angular frequency
• is the wave number
• The wave speed is given by:

From the given equation:

Substitute these values into the formula for wave speed:

Thus, the speed of the wave is 7.5 m/s.

To find the fundamental frequency of a vibrating wire, we use the formula for the fundamental frequency of a string fixed at both ends:

f =

Where:

• f is the fundamental frequency.
• L is the length of the wire (0.5 m).
• T is the tension in the wire (4 N).
• is the mass per unit length of the wire (0.0001 kg/m).

Substitute these values into the formula:

f =

f =

The square root of 40000 is 200, so:

f = 200 Hz

Hence, the fundamental frequency of the wire is 200 Hz.