A particle on a spring executes simple harmonic motion. when the particle is foubd at x=xm/2 the speed of the particle is

In simple harmonic motion (SHM), the behavior of a particle attached to a spring can be described using specific equations that relate its position, velocity, and acceleration. When we consider a particle at a position of \( x = \frac{x_m}{2} \), where \( x_m \) is the maximum displacement (or amplitude) of the motion, we can determine the speed of the particle at that point using the principles of energy conservation or the equations of motion for SHM.

Understanding Simple Harmonic Motion

In SHM, the total mechanical energy of the system remains constant and is the sum of kinetic energy (KE) and potential energy (PE). The potential energy stored in the spring when the particle is displaced from its equilibrium position can be expressed as:

• Potential Energy (PE): \( PE = \frac{1}{2} k x^2 \)
• Kinetic Energy (KE): \( KE = \frac{1}{2} mv^2 \)

Here, \( k \) is the spring constant, \( m \) is the mass of the particle, \( v \) is the speed of the particle, and \( x \) is the displacement from the equilibrium position.

Energy Conservation in SHM

The total mechanical energy (E) in SHM can be expressed as:

Total Energy (E): \( E = KE + PE = \frac{1}{2} k x_m^2 \)

At the maximum displacement \( x = x_m \), all the energy is potential, and at the equilibrium position \( x = 0 \), all the energy is kinetic. At any point in the motion, the total energy can be written as:

When the particle is at \( x = \frac{x_m}{2} \), we can calculate the potential energy:

Potential Energy at \( x = \frac{x_m}{2} \):

Substituting \( x = \frac{x_m}{2} \) into the potential energy formula:

\( PE = \frac{1}{2} k \left(\frac{x_m}{2}\right)^2 = \frac{1}{2} k \frac{x_m^2}{4} = \frac{1}{8} k x_m^2 \)

Now, we can find the kinetic energy at this position using the conservation of energy:

Kinetic Energy at \( x = \frac{x_m}{2} \):

Since the total energy \( E = \frac{1}{2} k x_m^2 \), we have:

\( KE = E - PE = \frac{1}{2} k x_m^2 - \frac{1}{8} k x_m^2 = \frac{4}{8} k x_m^2 - \frac{1}{8} k x_m^2 = \frac{3}{8} k x_m^2 \)

Finding the Speed

Now that we have the kinetic energy, we can use it to find the speed of the particle:

Setting the kinetic energy equal to \( \frac{1}{2} mv^2 \):

\( \frac{1}{2} mv^2 = \frac{3}{8} k x_m^2 \)

Solving for \( v \):

\( mv^2 = \frac{3}{4} k x_m^2 \)

\( v^2 = \frac{3}{4} \frac{k}{m} x_m^2 \)

\( v = \sqrt{\frac{3}{4} \frac{k}{m}} x_m \)

Thus, the speed of the particle when it is at \( x = \frac{x_m}{2} \) can be expressed as:

Final Speed Formula: \( v = \frac{\sqrt{3}}{2} \sqrt{\frac{k}{m}} x_m \)

This result shows that the speed of the particle at half the maximum displacement is proportional to the amplitude and depends on the spring constant and mass of the particle. This relationship is fundamental in understanding the dynamics of simple harmonic motion.