A horizontal spring is lying on a frictionless surface. One end of the spring is attaches to a wall while the other end is connected to a movable object. The spring and object are compressed by 0.074 m, released from rest, and subsequently oscillate back and forth with an angular frequency of 14.2 rad/s. What is the speed of the object at the instant when the spring is stretched by 0.041 m relative to its unstrained length?

To find the speed of the object at the instant when the spring is stretched by 0.041 m relative to its unstrained length, we can use the energy conservation principle in simple harmonic motion (SHM).

In SHM, the total mechanical energy of the system (spring and object) remains constant throughout the motion. The total mechanical energy is the sum of the potential energy (PE) stored in the spring and the kinetic energy (KE) of the object.

1. Find the potential energy of the spring when it is stretched by 0.041 m:
The potential energy of a spring is given by the equation PE = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position.

Since the spring is stretched by 0.074 m initially, and we need to find the potential energy when it is stretched by 0.041 m relative to its unstrained length, we subtract the two displacements:

Δx = 0.074 m - 0.041 m = 0.033 m

Now, we can calculate the potential energy:
PE = (1/2)k(Δx)^2

2. Calculate the potential energy when the spring is stretched by 0.033 m:
Substitute the given values:
PE = (1/2)k(0.033)^2

3. Find the kinetic energy of the object:
The kinetic energy of an object in SHM is given by the equation KE = (1/2)mv^2, where m is the mass of the object and v is its velocity.

Since the mass of the object is not given in the question, we cannot directly calculate the value of kinetic energy. However, we can express the kinetic energy in terms of potential energy using the energy conservation principle:

Total mechanical energy (E) = PE + KE

As mentioned earlier, the total mechanical energy remains constant throughout the motion. Therefore, we can write:

E = PE + KE_initial (initial position)
E = PE + KE_final (final position)

From this equation, we can solve for KE_final, which is the value we need.

4. Solving for KE_final:
Rearrange the equation:
KE_final = E - PE

Since the total mechanical energy (E) remains constant, we can assume that the total mechanical energy is equal to the potential energy initially (KE_initial = 0):
E = PE_initial

Hence, KE_final = E - PE_initial

Substitute the given values:
KE_final = (1/2)k(0.074)^2 - (1/2)k(0.033)^2

5. Calculate the speed of the object:
We can now find the speed (v) of the object using the kinetic energy equation:
KE_final = (1/2)mv^2

Rearrange the equation to solve for v:
v = sqrt(2 * KE_final / m)

Substitute the calculated KE_final value and the mass of the object (if given) into the equation to find the speed of the object.