A block of mass m = 350 g is released from rest and slides down a frictionless track of height h = 58.9 cm. At the bottom of the track the block slides freely along a horizontal table until it hits a spring attached to a heavy, immovable wall. The spring compressed by 2.35 cm at the maximum compression. What is the value of the spring constant k?
To find the value of the spring constant, we can use the conservation of mechanical energy.
First, we need to determine the potential energy of the block at the top of the track and the kinetic energy of the block at the bottom of the track.
At the top of the track, the block only has potential energy which can be calculated as:
Potential energy = mass * gravity * height
= 0.35 kg * 9.8 m/s^2 * 0.589 m
= 2.045 J
At the bottom of the track, the block only has kinetic energy which can be calculated as:
Kinetic energy = 0.5 * mass * velocity^2
Since the block slides freely along a horizontal table, there is no friction, so no energy is lost. This means that the kinetic energy at the bottom of the track will be equal to the potential energy at the top of the track.
Therefore, we can set the potential energy equal to the kinetic energy:
2.045 J = 0.5 * 0.35 kg * velocity^2
Now, we can solve for the velocity:
velocity^2 = (2.045 J) / (0.5 * 0.35 kg)
= 11.685 m^2/s^2
velocity = √(11.685) m/s
velocity = 3.417 m/s
When the block hits the spring at the maximum compression, all the kinetic energy is converted into potential energy stored in the compressed spring.
Potential energy of the spring = 0.5 * k * x^2
where k is the spring constant and x is the compression of the spring (2.35 cm = 0.0235 m).
Substituting the values, we have:
2.045 J = 0.5 * k * (0.0235 m)^2
Now, solve for k:
k = (2.045 J) / (0.5 * (0.0235 m)^2)
k = 7396.71 N/m
Therefore, the value of the spring constant (k) is 7396.71 N/m.