A block with mass m = 2.00kg is placed against a spring on a frictionless incline with angle (30 degrees). (The block is not attached to the spring.) The spring with spring constant k = 19.6 N/cm, is compressed 20.0 cm and then released. a.) What is the elastic potential energy of the compressed spring? b.) What is the change inn the gravitational potential energy of the block-Earth system as the block moves from the release point to its highest point on the incline? c.) How far along the inline is the highest point from the release point?
To find the answers to these questions, we can use the principles of mechanical energy conservation. Mechanical energy is the sum of the kinetic energy, potential energy due to gravity, and potential energy due to elastic deformation of the spring.
a) To calculate the elastic potential energy of the compressed spring, we can use the formula:
Elastic Potential Energy = (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 19.6 N/cm or 196 N/m, and the displacement, x, is 20.0 cm or 0.2 m, we can substitute these values into the formula:
Elastic Potential Energy = (1/2) * 196 * (0.2)^2 = 1.96 J
So, the elastic potential energy of the compressed spring is 1.96 Joules.
b) The change in gravitational potential energy of the block-Earth system as the block moves from the release point to its highest point on the incline can be calculated using the formula:
Change in Gravitational Potential Energy = m * g * h
Where m is the mass of the block, g is the acceleration due to gravity, and h is the change in height.
Given that the angle of the incline is 30 degrees and the mass of the block, m, is 2.00 kg, we can find the change in height, h, as follows:
h = x * sin(angle)
Where x is the displacement along the incline, which is the distance from the release point to the highest point.
Given the angle, 30 degrees, and the displacement, we can calculate the change in height:
h = 0.2 m * sin(30 degrees) = 0.1 m
Now, substituting the values into the formula for change in gravitational potential energy:
Change in Gravitational Potential Energy = 2.00 kg * 9.8 m/s^2 * 0.1 m = 1.96 J
Therefore, the change in gravitational potential energy of the block-Earth system is also 1.96 Joules.
c) To find the distance along the incline from the release point to the highest point, we can use the trigonometric relationships between the angle, the height change, and the distance along the incline.
In this case, we know that the displacement, x, along the incline is related to the change in height, h, by the formula:
x = h / sin(angle)
Using the values we already calculated, we can find the distance along the incline:
x = 0.1 m / sin(30 degrees) ≈ 0.2 m
So, the highest point on the incline is approximately 0.2 meters away from the release point.
In summary:
a) The elastic potential energy of the compressed spring is 1.96 Joules.
b) The change in gravitational potential energy of the block-Earth system is 1.96 Joules.
c) The highest point on the incline is approximately 0.2 meters away from the release point.