A 0.280 kg block on a vertical spring with a spring constant of 4.12 × 10^
3 N/m is pushed downward, compressing the spring 0.0600 m.
When released, the block leaves the spring and travels upward vertically.
The acceleration of gravity is 9.81 m/s^
2. How high does it rise above the point of release?
Answer in units of m
Just giving the equation to solve the problem would be nice too!
The potential energy of the compressed spring at release equals the potential energy gained at the maximum distance above the release point. Let that distnce be X and the initial spring compression be d = 0.06 m
(1/2) k d^2 = m g X
k is the spring constant, which you know is 4120 N/m.
m = 0.28 kg
Solve for X
To find the height that the block rises above the point of release, we can use the principle of conservation of energy.
The total mechanical energy of the block-spring system is conserved, meaning the initial potential energy stored in the compressed spring is converted into gravitational potential energy as the block rises. At the highest point, all the initial potential energy is converted into gravitational potential energy.
To solve this problem, we need to calculate the initial potential energy stored in the spring when it is compressed and then equate it to the final gravitational potential energy when the block reaches its highest point.
The initial potential energy stored in the compressed spring can be calculated using the formula:
U_spring = (1/2) * k * x^2
where U_spring is the potential energy stored in the spring, k is the spring constant, and x is the displacement of the spring from its equilibrium position.
Plugging in the values:
k = 4.12 × 10^3 N/m
x = 0.0600 m
U_spring = (1/2) * (4.12 × 10^3 N/m) * (0.0600 m)^2
U_spring = 0.07416 J
Next, we equate this initial potential energy to the final gravitational potential energy at the highest point. The gravitational potential energy can be calculated using the formula:
U_gravity = m * g * h
where U_gravity is the gravitational potential energy, m is the mass of the block, g is the acceleration due to gravity, and h is the height above the point of release.
Plugging in the values:
m = 0.280 kg
g = 9.81 m/s^2
0.07416 J = (0.280 kg) * (9.81 m/s^2) * h
Now, we can solve for h:
h = 0.07416 J / (0.280 kg * 9.81 m/s^2)
h ≈ 0.0275 m
Therefore, the block rises approximately 0.0275 meters above the point of release.