A mass sitting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. 5.5 J of work is required to compressed the spring by 0.16 m. If the mass is released from rest with the spring compressed, it experiences a maximum acceleration of 27 m/s2. Find the spring constant? what is the mass?

To find the spring constant and the mass in this scenario, we can use the principles of work and energy, as well as Hooke's Law.

Step 1: Find the spring constant (k)
The work done to compress the spring is equal to the potential energy stored in the spring:

Work = Potential Energy

Given:
Work (W) = 5.5 J
Compression (x) = 0.16 m

The potential energy stored in a spring can be calculated using the formula:

Potential Energy = (1/2) * k * x^2

Substituting the given values into the equation:

5.5 J = (1/2) * k * (0.16 m)^2

Solving for k:

k = (2 * 5.5 J) / (0.16 m^2)
k = 68.75 N/m

Therefore, the spring constant is 68.75 N/m.

Step 2: Find the mass (m)

When the mass is released, it experiences a maximum acceleration. Using Hooke's Law and Newton's second law of motion, we can relate the maximum acceleration to the mass and spring constant:

a = (k / m) * x

Given:
Acceleration (a) = 27 m/s^2
Compression (x) = 0.16 m
Spring constant (k) = 68.75 N/m

Substituting the given values into the equation:

27 m/s^2 = (68.75 N/m) / m * (0.16 m)

Simplifying the equation:

27 m^2/s^2 = 68.75 N / m^2 * 0.16 m

27 m^2/s^2 = 11 N / m

Solving for the mass (m):

11 N / m = m * 27 m^2/s^2

m = (11 N / m) / (27 m^2/s^2)
m = 0.407 kg (approximately)

Therefore, the mass is approximately 0.407 kg.