A spring is mounted horizontally. A crate which has a mass of 8.5kg is pressed against the spring with a force of 350N. As a result the spring is compressed a distance of 82.0cm. The mass is then released and is allowed to slide along the horizontal, frictionless surface.
What will be the velocity of the crate after it has left the spring? (5.81m/s)
How far will the mass slide before it comes to a halt? (2.65m)
To determine the velocity of the crate after it has left the spring and the distance it will slide before coming to a halt, we can apply the principles of energy conservation.
1. Velocity of the crate after leaving the spring:
The potential energy stored in the spring when it is compressed is given by the formula: PE = (1/2)kx^2, where PE is the potential energy, k is the spring constant, and x is the displacement.
The work done by the crate in compressing the spring is equal to the potential energy stored in it. Therefore, the work done is given by the formula: Work = Force * Distance = PE.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. Thus, the work done on the spring is equal to the change in kinetic energy of the crate.
Therefore, we can write: (1/2)kx^2 = (1/2)mv^2, where m is the mass of the crate, and v is its velocity after leaving the spring.
Rearranging the equation, we have: v^2 = (k/m) * x^2.
Substituting the known values: k = 350 N/m, m = 8.5 kg, and x = 82.0 cm = 0.82 m, we can calculate the velocity as follows:
v^2 = (350 N/m / 8.5 kg) * (0.82 m)^2
v^2 = 34.1176
v = √34.1176
v ≈ 5.81 m/s
Therefore, the velocity of the crate after leaving the spring is approximately 5.81 m/s.
2. Distance the mass slides before coming to a halt:
To determine the distance the mass will slide before coming to a halt, we need to find the position where the kinetic energy of the crate becomes zero.
At the point where the crate comes to a halt, all its initial energy will be converted to potential energy (gravitational potential energy).
Using the work-energy theorem, we can equate the initial kinetic energy of the crate to its final potential energy: (1/2)mv^2 = mgh, where h is the height from the starting point (which is equal to the displacement from the spring).
Rearranging the equation, we have: h = (1/2g)(v^2), where g is the acceleration due to gravity.
Substituting the known values: v = 5.81 m/s and g = 9.8 m/s^2, we can calculate the distance as follows:
h = (1/2 * 9.8 m/s^2) * (5.81 m/s)^2
h ≈ 2.65 m
Therefore, the mass will slide approximately 2.65 m before coming to a halt.