A block with mass m = 2.00 kg is placed against a spring on a frictionless incline with angle θ=30.0° (see figure). The block is NOT attached to the spring. The spring, with spring constant k = 19.6 N/cm, is compressed 20.0 cm from its unloaded equilibrium length and then released.
(a) What is the elastic potential energy of the compressed spring?
(b) What is 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?
To solve this problem, we need to break it down into two parts: finding the elastic potential energy of the compressed spring and calculating the change in gravitational potential energy as the block moves from the release point to the highest point on the incline.
(a) Elastic Potential Energy of the Compressed Spring:
The elastic potential energy stored in a spring can be calculated using the formula:
Elastic Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement from equilibrium.
Given:
Spring constant, k = 19.6 N/cm
Displacement from equilibrium, x = 20.0 cm = 0.20 m
Substituting the values into the formula, we have:
Elastic Potential Energy = (1/2) * (19.6 N/cm) * (0.20 m)^2
Calculating this expression gives us the value of the elastic potential energy of the compressed spring.
(b) Change in Gravitational Potential Energy of the Block+Earth System:
The change in gravitational potential energy of an object moving vertically can be calculated using the formula:
Change in Gravitational Potential Energy = m * g * h
where m is the mass, g is the acceleration due to gravity, and h is the change in height.
Given:
Mass of the block, m = 2.00 kg
Angle of the incline, θ = 30.0°
To calculate h, we need to find the vertical displacement of the block from the release point to its highest point on the incline. This displacement can be found using trigonometry:
h = x * sin(θ)
where x is the horizontal displacement of the block.
From the figure, we can see that x is given by:
x = (20.0 cm) / cos(θ)
Now, substituting the values into the formula:
h = [(20.0 cm) / cos(θ)] * sin(θ)
Finally, we can calculate the change in gravitational potential energy using the formula and the calculated h value.
Change in Gravitational Potential Energy = (2.00 kg) * g * h
By substituting the value of g and the calculated value of h, the change in gravitational potential energy can be determined.