A 500 g block is released from rest and slides down a frictionless track that begins h = 1.70 m above the horizontal, as shown in Figure P13.56. At the bottom of the track, where the surface is horizontal, the block strikes and sticks to a light spring with a spring constant of 25.0 N/m. Find the maximum distance the spring is compressed.

m

(1/2) k X^2 = potential energy loss = M g H

Solve for X

To find the maximum distance the spring is compressed, we'll need to apply the principle of conservation of energy.

Step 1: Determine the potential energy at the top of the track.
- The potential energy (PE) of an object at a certain height is given by the equation PE = mgh, where m is the mass of the object, g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
- In this case, the mass (m) is given as 500 g, which is equal to 0.5 kg, and the height (h) is given as 1.70 m.
- Therefore, the potential energy at the top of the track is PE = (0.5 kg)(9.8 m/s^2)(1.70 m).

Step 2: Determine the maximum potential energy of the compressed spring.
- The potential energy of a spring (PE_spring) is given by the equation PE_spring = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position.
- In this case, the spring constant (k) is given as 25.0 N/m.
- Since the block sticks to the spring, its maximum displacement (x) will be the maximum compression of the spring.
- Therefore, the maximum potential energy of the compressed spring is PE_spring = (1/2)(25.0 N/m)(x)^2.

Step 3: Apply the principle of conservation of energy.
- According to the conservation of energy, the total energy at the top of the track (potential energy) is equal to the maximum potential energy of the compressed spring.
- Therefore, we can equate the two expressions from Step 1 and Step 2:
(0.5 kg)(9.8 m/s^2)(1.70 m) = (1/2)(25.0 N/m)(x)^2.

Step 4: Solve for x.
- Rearrange the equation and solve for x: x = sqrt((2 * (0.5 kg)(9.8 m/s^2)(1.70 m)) / (25.0 N/m)).

Step 5: Calculate the maximum distance the spring is compressed.
- Plug in the values and calculate: x ≈ 0.601 m.

Therefore, the maximum distance the spring is compressed is approximately 0.601 meters.