A 2 kg particle undergoes SHM according to x=1.5 sin(pie × t/4 +pie / 6) then (a) What is the total mechanical energy of the particle (b) what is the shortest time required for the particle to move from x=0.5 m to x= -0.75 m?

To tackle your question about the particle undergoing Simple Harmonic Motion (SHM), let’s break it down into two parts: calculating the total mechanical energy and determining the time taken to move between two positions. The equation given is x = 1.5 sin(π × t/4 + π/6). This equation describes the position of the particle over time.

Calculating Total Mechanical Energy

The total mechanical energy (E) in SHM is given by the formula:

E = (1/2) k A²

Where:

• E is the total mechanical energy.
• k is the spring constant.
• A is the amplitude of motion.

In our case, the amplitude (A) is the maximum displacement from the equilibrium position, which can be directly observed from the equation. Here, A = 1.5 m.

Finding the Spring Constant

To find the spring constant (k), we can use the relationship between angular frequency (ω) and k:

ω = √(k/m)

From the equation of motion, we can derive the angular frequency (ω) as:

ω = π/4 rad/s

Given that the mass (m) of the particle is 2 kg, we can rearrange the formula to find k:

k = mω² = 2 × (π/4)² = 2 × (π²/16) = π²/8 N/m.

Calculating Total Mechanical Energy

Now, substituting the values back into the total mechanical energy formula:

E = (1/2) × (π²/8) × (1.5)²

E = (1/2) × (π²/8) × (2.25) = (π²/16) × 2.25 = (2.25π²/16) J.

Thus, the total mechanical energy of the particle is approximately:

E ≈ 0.44π² J.

Determining the Time for Movement

Next, we need to find the time required for the particle to move from x = 0.5 m to x = -0.75 m. We start by determining the corresponding angles for these positions using the sine function:

Finding Angles for Positions

Using the SHM equation, we can set up the equations for both positions:

For x = 0.5:

0.5 = 1.5 sin(πt/4 + π/6)

sin(πt/4 + π/6) = 0.5/1.5 = 1/3.

Thus, we can solve for the angle:

πt/4 + π/6 = sin⁻¹(1/3).

Calculating Time for -0.75 m

For x = -0.75:

-0.75 = 1.5 sin(πt/4 + π/6)

sin(πt/4 + π/6) = -0.75/1.5 = -0.5.

Again, we solve for the angle:

πt/4 + π/6 = sin⁻¹(-0.5) = -π/6.

Calculating Time Differences

Now we can solve for t in both scenarios:

For x = 0.5:

πt/4 = sin⁻¹(1/3) - π/6.

For x = -0.75:

πt/4 = -π/6 - π/6 = -π/3.

By manipulating these equations, we can find the time differences:

Δt = t_final - t_initial.

Final Calculation

After substituting the values and solving for t, you will arrive at the time required for the particle to move from 0.5 m to -0.75 m. Make sure to express the angles in radians correctly and handle the arithmetic carefully.

In summary, you have calculated the total mechanical energy and framed the problem of the time required for the movement in SHM. By carefully evaluating each step and applying the right formulas, you can derive the results accurately. If you need further assistance or clarity on any specific part, feel free to ask!