A block of mass 12.0 kg is initially sitting at rest on a 30.0o incline. There is a spring mounted to the ramp with spring constant k = 13500 N/m. The block is released to slide down the incline and comes to a stop momentarily after the spring has been compressed a distance of 5.5 cm. a) How far along the incline has the block moved from the point at which it was released to this point at which it come to rest momentarily? b) What is the speed of the block as it first touches the spring?

To solve this problem, we can use the principle of conservation of mechanical energy and Hooke's Law.

a) To find the distance along the incline that the block has moved, we can equate the potential energy at the starting point to the potential energy at the point where the block comes to rest momentarily.

The potential energy at the starting point is given by:
PE_start = m * g * h_start
where m is the mass of the block, g is the acceleration due to gravity, and h_start is the height of the starting point (which we can calculate using the inclined plane).

The potential energy at the point where the block comes to rest momentarily is given by:
PE_rest = m * g * h_rest
where h_rest is the height of the rest point (which we can calculate using the compressed distance of the spring).

Since the block comes to rest momentarily, we can equate the potential energies:
PE_start = PE_rest

Next, we need to calculate the height of the starting point (h_start) and the height of the rest point (h_rest) along the incline.

To find h_start, we can use trigonometry:
h_start = d * sin(theta)
where d is the distance along the incline and theta is the angle of the incline (given as 30.0 degrees).

To find h_rest, we need to know the compression of the spring (x) and the spring constant (k). We can use Hooke's Law:
F_spring = k * x
where F_spring is the force exerted by the spring.

The force exerted by the spring is equal to the weight component parallel to the incline. We can calculate it using the weight of the block:
F_weight = m * g
F_parallel = F_weight * sin(theta)

Now, we can set up the equation for Hooke's Law:
F_spring = F_parallel
k * x = m * g * sin(theta)
x = (m * g * sin(theta)) / k

Substituting the calculated values of x, m, g, and theta, we can find h_rest.

Finally, we can substitute the values of h_start and h_rest into the equation PE_start = PE_rest to find the distance along the incline that the block has moved (d).

b) To find the speed of the block as it first touches the spring, we can use the principle of conservation of mechanical energy.

The initial mechanical energy (E_initial) when the block is released is given by:
E_initial = PE_start + KE_start
where KE_start is the initial kinetic energy.

The final mechanical energy (E_final) when the block comes to rest momentarily is given by:
E_final = PE_rest + KE_rest
where KE_rest is the kinetic energy at the rest point (which is zero since the block comes to rest).

Since the mechanical energy is conserved, we can equate E_initial and E_final:
PE_start + KE_start = PE_rest

Next, we need to calculate KE_start. Since the block is released at rest, KE_start is zero.

Substituting the values of PE_start and PE_rest, we can solve for KE_rest.

Finally, we can use the equation for kinetic energy to find the speed of the block as it first touches the spring:
KE_start = (1/2) * m * v^2
where v is the speed of the block.

Substituting the value of KE_start, we can solve for v.

By following these steps, we can find the answers to both parts (a) and (b) of the problem.