A 0.250 kg block on a vertical spring with a spring constant of 4.09 × 103 N/m is pushed downward, compressing the spring 0.0900 m. When released, the block leaves the spring and travels upward vertically.
The acceleration of gravity is 9.81 m/s2 .
How high does it rise above the point of release?
Answer in units of m
To find the height the block rises above the point of release, we need to use conservation of mechanical energy.
First, let's determine the initial potential energy (U_initial) of the block when it is compressed in the spring. The potential energy of a spring is given by the formula: U_initial = (1/2) * k * x^2, where k is the spring constant and x is the displacement from the equilibrium position.
Given that the spring constant (k) is 4.09 × 10^3 N/m and the displacement (x) is 0.0900 m, we can calculate the initial potential energy:
U_initial = (1/2) * (4.09 × 10^3 N/m) * (0.0900 m)^2
Next, we know that the initial potential energy (U_initial) is converted into gravitational potential energy (U_final) as the block rises. Therefore, the height (h) that the block rises above the point of release is equal to U_final divided by the product of the mass (m) and acceleration due to gravity (g).
Using the equation for gravitational potential energy: U_final = m * g * h, we can rearrange the equation to solve for h:
h = U_final / (m * g)
Given that the mass (m) is 0.250 kg and the acceleration due to gravity (g) is 9.81 m/s^2, we can calculate the height:
h = U_final / (0.250 kg * 9.81 m/s^2)
So, in order to find the answer to the question, we need to calculate U_final and substitute the values into the equation above.
To solve this problem, we can use the principle of conservation of mechanical energy. The mechanical energy of the block-spring system is conserved as long as no non-conservative forces act on it.
Let's calculate the potential energy stored in the compressed spring using the formula:
Potential energy (U) = (1/2) * k * x^2
where k is the spring constant and x is the compression of the spring.
Given:
Mass of the block (m) = 0.250 kg
Spring constant (k) = 4.09 × 10^3 N/m
Compression of the spring (x) = 0.0900 m
Plugging in the values, we get:
Potential energy (U) = (1/2) * (4.09 × 10^3 N/m) * (0.0900 m)^2
Calculating U, we find:
U ≈ 16.6 J
Now, when the block is released, it will have both potential energy and kinetic energy. As the block travels upward against gravity, its potential energy will convert to kinetic energy and vice versa.
At the highest point, when the block momentarily comes to a rest, all its energy will be in the form of potential energy.
Using the principle of conservation of mechanical energy, we can equate the potential energy at the highest point to the initial potential energy:
Potential energy at highest point = Potential energy initially
m * g * h = U
where g is the acceleration due to gravity and h is the maximum height reached by the block.
Since we need to find h, let's rearrange the equation:
h = U / (m * g)
Plugging in the values, we get:
h = (16.6 J) / (0.250 kg * 9.81 m/s^2)
Calculating h, we find:
h ≈ 6.70 m
Therefore, the block will rise approximately 6.70 meters above the point of release.