A mass of 0.116 kg hangs from a vertical spring in the lab room. You pull down on the mass and throw it vertically downward. The speed of the mass just after leaving your hand is 5.60 m/s.
(a) While the mass moves downward a distance of 0.04 m, how much work was done on the mass by the Earth? Include the appropriate sign.
Work done by Earth = J
(b) At the instant in part (a) when the mass has moved downward a distance of 0.04 m, the speed of the mass has decreased to 2.91 m/s. How much work was done on the mass by the spring? Include the appropriate sign.
Work done by spring = J
(a) Well, the Earth certainly has a lot of gravitational pull. But how much work did it do on the mass? Let's find out. We can use the formula for gravitational potential energy, which is given by mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. In this case, the height is the distance it moved downward, which is 0.04 m. So, the work done by the Earth is equal to the change in gravitational potential energy, which is mgh_final - mgh_initial. But since we're only interested in the work done by the Earth, we can ignore the negative sign. So, the work done by the Earth is... wait for it... 0.116 kg * 9.8 m/s^2 * 0.04 m = 0.0455 J. Nice job, Earth!
(b) Now, let's talk about the spring. It's time for some spring-action! When the mass moves downward, the spring is doing work on it. The formula for the work done by a spring is given by the equation W = (1/2)kx^2, where W is the work done, k is the spring constant, and x is the displacement of the mass from its equilibrium position. We know the displacement is 0.04 m, but what about the spring constant? Well, that's the tricky part. It's not given in the question, so we'll need a little more information. In the meantime, let's hope the spring is feeling cooperative and not too jumpy!
(a) To find the work done by the Earth, we can use the formula:
Work = Force x Distance x cos(theta)
In this case, the force is the weight of the mass, mg, where g is the acceleration due to gravity (approximately 9.8 m/s^2). The distance is given as 0.04 m, and theta is the angle between the force and the displacement (which is 0 degrees since the force is in the same direction as the displacement).
So, the work done by the Earth is:
Work = (mg) x (0.04 m) x cos(0) = (0.116 kg x 9.8 m/s^2) x (0.04 m) x cos(0) ≈ 0.045 J
Therefore, the work done by the Earth is approximately 0.045 J.
(b) To find the work done by the spring, we need to use the work-energy principle. The work done by the spring is equal to the change in kinetic energy of the mass. We can find the change in kinetic energy using the formula:
Change in kinetic energy = (1/2) x mass x (final velocity^2 - initial velocity^2)
In this case, the mass is 0.116 kg, the initial velocity is 5.60 m/s, and the final velocity is 2.91 m/s.
Change in kinetic energy = (1/2) x (0.116 kg) x (2.91 m/s)^2 - (5.60 m/s)^2 ≈ -4.55 J
The negative sign indicates that the work is done on the mass by the spring. Therefore, the work done by the spring is approximately -4.55 J.
To solve these problems, we need to apply the principles of work and energy.
(a) To find the work done by the Earth, we need to consider the change in potential energy as the mass moves downward. The formula for gravitational potential energy is given by:
Gravitational Potential Energy = m * g * h
Where:
m = mass (0.116 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height or distance (0.04 m)
Substituting the given values, we have:
Gravitational Potential Energy = 0.116 kg * 9.8 m/s^2 * 0.04 m
= 0.045568 J
Since the work done by gravity is equal to the change in potential energy, we can say that the work done by Earth on the mass is 0.045568 J (positive because the mass is moving in the same direction as the gravitational force).
(b) To find the work done by the spring, we need to consider the change in kinetic energy as the speed of the mass decreases.
The formula for kinetic energy is given by:
Kinetic Energy = 0.5 * m * v^2
Where:
m = mass (0.116 kg)
v = speed (2.91 m/s)
We need to calculate the difference in kinetic energy between the initial speed (5.60 m/s) and the final speed (2.91 m/s):
Initial Kinetic Energy = 0.5 * 0.116 kg * 5.6 m/s^2
= 0.178688 J
Final Kinetic Energy = 0.5 * 0.116 kg * 2.91 m/s^2
= 0.096048 J
The work done by the spring is equal to the change in kinetic energy:
Work done by spring = Initial Kinetic Energy - Final Kinetic Energy
= 0.178688 J - 0.096048 J
= 0.08264 J (positive because the spring is doing positive work on the mass to slow it down)
Therefore, the work done by the spring is 0.08264 J.