A 710 gram block is released from a height h above an ideal spring of spring constant 400 N/m. The block sticks to the spring, and comes momentarily to rest after compressing the spring by an amount of 18.7 cm.

DropOnSpring
(a) What is the value of the initial height h?
Correct: Your answer is correct.

(b) What is the height of the block when it again comes to rest momentarily after it bounces back up? Assume that the block remains connected to the spring, and measure the height relative to the equilibrium position of the spring.

To find the value of the initial height h, we can use conservation of mechanical energy.

The potential energy of the block at height h is given by:

PE = mgh

where m is the mass of the block (0.710 kg or 710 g), g is the acceleration due to gravity (9.8 m/s^2), and h is the initial height.

Since the block comes to rest momentarily after compressing the spring by 18.7 cm (0.187 m) and sticks to the spring, all of the initial potential energy is converted into spring potential energy at the moment of rest. The spring potential energy is given by:

PE_spring = (1/2)kx^2

where k is the spring constant (400 N/m) and x is the compression of the spring (0.187 m).

Equating the initial potential energy to the spring potential energy, we have:

mgh = (1/2)kx^2

Substituting the given values:

0.710 kg * 9.8 m/s^2 * h = (1/2) * 400 N/m * (0.187 m)^2

Simplifying and solving for h, we get:

h = (1/2) * (0.187 m)^2 * (400 N/m) / (0.710 kg * 9.8 m/s^2)

h ≈ 0.169 m

So, the value of the initial height h is approximately 0.169 meters.

To find the height of the block when it again comes to rest momentarily after it bounces back up, we need to consider the gravitational potential energy and the spring potential energy.

At the equilibrium position of the spring (when it is neither compressed nor stretched), the gravitational potential energy is zero and all the energy is in the form of spring potential energy. So, at this point, the height of the block relative to the equilibrium position of the spring is 0.

When the block bounces back up, the spring potential energy is converted back into gravitational potential energy. The maximum height reached by the block can be found using the conservation of mechanical energy.

We can use the following equation:

PE_spring = mgh_max

where h_max is the maximum height we need to find.

mgh_max = (1/2)kx^2

Substituting the given values:

0.710 kg * 9.8 m/s^2 * h_max = (1/2) * 400 N/m * (0.187 m)^2

Simplifying and solving for h_max, we get:

h_max = (1/2) * (0.187 m)^2 * (400 N/m) / (0.710 kg * 9.8 m/s^2)

h_max ≈ 0.169 m

So, the height of the block when it again comes to rest momentarily after it bounces back up is approximately 0.169 meters relative to the equilibrium position of the spring.

To find the value of the initial height of the block, we can use the principle of conservation of energy. The potential energy of the block at the initial height is converted into the spring potential energy when it comes to rest momentarily after compressing the spring.

(a)
The potential energy of the block at height h is given by the formula:

Potential Energy = m * g * h

Where:
m = mass of the block (710 grams = 0.71 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = initial height

The spring potential energy when the block comes to rest is given by:

Spring Potential Energy = 0.5 * k * x^2

Where:
k = spring constant (400 N/m)
x = amount of compression (18.7 cm = 0.187 m)

Equating the two energies to each other:

m * g * h = 0.5 * k * x^2

Substituting the given values:

0.71 kg * 9.8 m/s^2 * h = 0.5 * 400 N/m * (0.187 m)^2

Simplifying the equation:

h = (0.5 * 400 N/m * (0.187 m)^2) / (0.71 kg * 9.8 m/s^2)

Calculating the value of h will give us the answer.

(b)
To find the height of the block when it again comes to rest momentarily after bouncing back up, we can use the same principle of conservation of energy. The potential energy of the spring is converted back into the potential energy of the block at the maximum height.

This time, the potential energy of the block at the maximum height is given by:

Potential Energy = m * g * h_max

Where:
h_max = maximum height attained by the block after bouncing back up.

Equating the two energies to each other:

0.5 * k * x^2 = m * g * h_max

Substituting the known values:

0.5 * 400 N/m * (0.187 m)^2 = 0.71 kg * 9.8 m/s^2 * h_max

Rearranging the equation to solve for h_max:

h_max = (0.5 * 400 N/m * (0.187 m)^2) / (0.71 kg * 9.8 m/s^2)

Calculating the value of h_max will give us the answer.