A 3.30 kg object is hanging from the end of a vertical spring. The spring constant is 48.0 N/m. The object is pulled 0.200 m downward and released from rest. Complete the table below by calculating the translational kinetic energy, the gravitational potential energy, the elastic potential energy, and the total mechanical energy E for each of the vertical positions indicated. The vertical positions h indicate distances above the point of release, where h = 0.

Find KE, PE-gravity, PE-elastic and E for each h (0, 0.200, 0.400)

The spring is already stretched by the weight before is is pulled. That amount of "equilibrium" stretch is

Xe = M g/k = 0.674 m. Then the spring is pulled downward another 0.200 m for a total stretch of X = 0.874 m at the point of release. The kinetic energy is zero at that point, and the spring potential energy is (1/2)kX^2 = 18.33 J . Relative to the the point of release, the position is then h = 0. Call the gravitational potential energy zero at that point also.

The total energy of the system is the spring energy of 18.33 J
At h = 0.2, X = 0.674 m
At h = 0.4, X = 0.474 m
gravitational PE (gPE) = M g h
spring potential energy = (1/2) k X^2
kinetic energy = 18.33 J - KE - gPE

You complete the table

To find the kinetic energy (KE), gravitational potential energy (PE-gravity), elastic potential energy (PE-elastic), and total mechanical energy (E) for each position, we can use the following formulas:

1. Kinetic Energy (KE): KE = 1/2 * m * v^2
2. Gravitational Potential Energy (PE-gravity): PE-gravity = m * g * h
3. Elastic Potential Energy (PE-elastic): PE-elastic = 1/2 * k * x^2
4. Total Mechanical Energy (E): E = KE + PE-gravity + PE-elastic

Given:
Mass (m) = 3.30 kg
Spring Constant (k) = 48.0 N/m
Displacement (x) = 0.200 m
Height (h) = 0, 0.200, 0.400
Gravity (g) = 9.8 m/s^2

Let's calculate the values for each position:

For h = 0:
1. KE = 0 (as the object is released from rest)
2. PE-gravity = m * g * h = 3.30 kg * 9.8 m/s^2 * 0 m = 0 J
3. PE-elastic = 1/2 * k * x^2 = 1/2 * 48.0 N/m * (0.200 m)^2 = 0.96 J
4. E = KE + PE-gravity + PE-elastic = 0 J + 0 J + 0.96 J = 0.96 J

For h = 0.200:
1. KE = 1/2 * m * v^2 = 1/2 * 3.30 kg * (2gh)^2 = 1/2 * 3.30 kg * (2 * 9.8 m/s^2 * 0.2 m)^2 = 0.6464 J
2. PE-gravity = m * g * h = 3.30 kg * 9.8 m/s^2 * 0.200 m = 6.468 J
3. PE-elastic = 1/2 * k * x^2 = 1/2 * 48.0 N/m * (0.200 m)^2 = 0.96 J
4. E = KE + PE-gravity + PE-elastic = 0.6464 J + 6.468 J + 0.96 J = 8.0744 J

For h = 0.400:
1. KE = 1/2 * m * v^2 = 1/2 * 3.30 kg * (2gh)^2 = 1/2 * 3.30 kg * (2 * 9.8 m/s^2 * 0.4 m)^2 = 2.5856 J
2. PE-gravity = m * g * h = 3.30 kg * 9.8 m/s^2 * 0.400 m = 12.936 J
3. PE-elastic = 1/2 * k * x^2 = 1/2 * 48.0 N/m * (0.200 m)^2 = 0.96 J
4. E = KE + PE-gravity + PE-elastic = 2.5856 J + 12.936 J + 0.96 J = 16.4826 J

Now, let's summarize the values in the table:

| Position (h) | KE (J) | PE-gravity (J) | PE-elastic (J) | E (J) |
|--------------|--------|----------------|----------------|----------|
| 0 | 0 | 0 | 0.96 | 0.96 |
| 0.200 | 0.6464 | 6.468 | 0.96 | 8.0744 |
| 0.400 | 2.5856 | 12.936 | 0.96 | 16.4826 |

To find the kinetic energy (KE), gravitational potential energy (PE-gravity), elastic potential energy (PE-elastic), and total mechanical energy (E) for each vertical position (h), you will need to use the appropriate formulas and equations.

1. KE (Translational Kinetic Energy):
The formula for kinetic energy is given by:
KE = (1/2) * m * v^2
where m is the mass of the object and v is its velocity.

For h = 0 (point of release):
Since the object is released from rest, its velocity is 0. Therefore, the kinetic energy at this position is also 0.

For h = 0.200 m:
To determine the velocity at this position, you can use the concept of conservation of mechanical energy. The total mechanical energy (E) remains constant in the absence of external forces.

E = PE-gravity + PE-elastic + KE

At point of release (h = 0), the total mechanical energy is given by:
E = m * g * h + (1/2) * k * h^2
Substituting the given values:
E = (3.30 kg) * (9.8 m/s^2) * 0 + (1/2) * (48.0 N/m) * (0)^2 = 0 + 0 = 0

Using the conservation of mechanical energy, we can equate the total mechanical energy at the point of release (E = 0) to the sum of the potential and kinetic energy at any other position:

E = m * g * h + (1/2) * k * h^2
0 = (3.30 kg) * (9.8 m/s^2) * (0.200 m) + (1/2) * (48.0 N/m) * (0.200 m)^2

Now we can solve for the velocity squared (v^2):
(1/2) * (48.0 N/m) * (0.200 m)^2 = (3.30 kg) * (9.8 m/s^2) * (0.200 m) * v^2
v^2 = (1/2) * (48.0 N/m) * (0.200 m)

Finally, we can calculate the kinetic energy at h = 0.200 m:
KE = (1/2) * m * v^2 = (1/2) * (3.30 kg) * [(1/2) * (48.0 N/m) * (0.200 m)]

2. PE-gravity (Gravitational Potential Energy):
The formula for gravitational potential energy is:
PE-gravity = m * g * h
where m is the mass, g is the acceleration due to gravity, and h is the vertical position.

For h = 0 (point of release), the gravitational potential energy is:
PE-gravity = (3.30 kg) * (9.8 m/s^2) * 0 = 0

For h = 0.200 m, the gravitational potential energy is:
PE-gravity = (3.30 kg) * (9.8 m/s^2) * (0.200 m)

3. PE-elastic (Elastic Potential Energy):
The formula for elastic potential energy stored in a spring is given by Hooke's Law:
PE-elastic = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.

For h = 0 (point of release), there is no displacement from the equilibrium position of the spring. Therefore, the elastic potential energy is:
PE-elastic = (1/2) * (48.0 N/m) * (0)^2 = 0

For h = 0.200 m, the displacement from the equilibrium position is x = h, and the elastic potential energy is:
PE-elastic = (1/2) * (48.0 N/m) * (0.200 m)^2

4. Total Mechanical Energy (E):
The total mechanical energy (E) is the sum of the kinetic energy (KE), gravitational potential energy (PE-gravity), and elastic potential energy (PE-elastic).

For h = 0 (point of release), the total mechanical energy is:
E = 0 + 0 + 0 = 0

For h = 0.200 m:
E = KE + PE-gravity + PE-elastic
E = KE + m * g * h + (1/2) * k * h^2

Now that you have the formulas and equations to determine KE, PE-gravity, PE-elastic, and E for each vertical position (h), you can substitute the given values into the appropriate equations to calculate the values.