A 0.340 kg block on a vertical spring with a spring constant of 4.09 × 103 N/m is pushed downward, compressing the spring 0.0700 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
spring energy= change gravational PE
1/2 k x^2 = mgh solve for h.
To determine the height the block rises above the point of release, we can use the principle of conservation of mechanical energy.
The potential energy stored in the compressed spring is given by the equation:
PE_spring = 0.5 * k * x^2
where PE_spring is the potential energy stored in the spring, k is the spring constant, and x is the displacement of the spring.
Given:
Mass of the block (m) = 0.340 kg
Spring constant (k) = 4.09 × 10^3 N/m
Displacement of the spring (x) = 0.0700 m
Substitute the values into the equation:
PE_spring = 0.5 * 4.09 × 10^3 N/m * (0.0700 m)^2
PE_spring = 0.5 * 4.09 × 10^3 N/m * 0.0049 m^2
PE_spring = 10.019 N·m
The potential energy stored in the compressed spring will be converted into gravitational potential energy when the block rises above the point of release.
The gravitational potential energy (PE_gravity) is given by the equation:
PE_gravity = m * g * h
where m is the mass of the block, g is the acceleration due to gravity, and h is the height above the point of release.
We can set the potential energy of the spring equal to the potential energy due to gravity and solve for h:
PE_spring = PE_gravity
10.019 N·m = 0.340 kg * 9.81 m/s^2 * h
10.019 N·m = 3.33654 N * h
Dividing both sides by 3.33654 N:
h = 10.019 N·m / 3.33654 N
h ≈ 3.009 m
Therefore, the block will rise approximately 3.009 meters above the point of release.
To find the height the block rises above the point of release, we can use the conservation of mechanical energy.
When the block is released, it will have both potential energy and kinetic energy. The potential energy initially stored in the compressed spring will convert into kinetic energy when the block leaves the spring, and then into potential energy as the block rises.
The potential energy stored in the spring is given by the equation:
PE_spring = (1/2) * k * (x^2)
Where k is the spring constant and x is the compression or extension of the spring. In this case, x = 0.0700 m.
PE_spring = (1/2) * (4.09 * 10^3 N/m) * (0.0700 m)^2
= 1.897 J (joules)
To find the maximum height the block rises, we can equate the potential energy at the highest point to the initial potential energy stored in the spring.
PE_gravitational = PE_spring
At the highest point, all potential energy will be in the form of gravitational potential energy:
PE_gravitational = m * g * h
Where m is the mass of the block, g is the acceleration due to gravity, and h is the height above the point of release.
We can rearrange the equation to solve for h:
h = PE_gravitational / (m * g)
Plugging in the values:
h = 1.897 J / (0.340 kg * 9.81 m/s^2)
= 0.56 m
Therefore, the block rises 0.56 meters above the point of release.