A box slides down a frictionless, 10-meter, 30o incline. Calculate the gravitational potential energy at both the beginning and end of the motion. Calculate the work done by GRAVITY as the box slides down the incline. Compare this to the overall change in gravitational potential energy. What conclusions can you draw from this?
2. Potential Energy is only stored in a system when... (How to finish it off)
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When a 5 kg box is lifted 3 meters from the ground, calculate the gravitational potential energy stored in the box. Calculate the work done by a Force to lift the box at a constant speed. Compare this answer to the amount of gravitational potential energy stored in the box. What conclusions can you draw from this? What would happen if you did more work lifting the box than what was stored as gravitational potential energy?
ΔUg= mg Δy
ΔUg= (5kg)(-9.8 m/s^2)(3m)
ΔUg= -147J ( gravitational potential energy)
Calculate work:
W= F * D
= (5kg)(-9.8 m/s^2)
=-49 J
See prev post
For the first question:
To calculate the gravitational potential energy at the beginning and end of the motion, we can use the equation:
Potential Energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
At the beginning of the motion, the height is 0, so the gravitational potential energy is 0.
At the end of the motion, the height is 10 meters. Assuming the mass of the box is m, the gravitational potential energy can be calculated as follows:
PE_end = m * g * h
= m * 9.8 m/s^2 * 10 m
= 98 m J, where m is the mass of the box in kilograms.
Now, let's calculate the work done by gravity as the box slides down the incline. The work done by gravity can be calculated using the formula:
Work (W) = force (F) * displacement (d) * cos(theta)
Since the incline is frictionless, the only force acting on the box is its weight, which is given by m * g, where m is the mass of the box and g is the acceleration due to gravity.
The displacement of the box down the incline is 10 meters, and the angle between the force of gravity and the displacement is 30 degrees.
Therefore, the work done by gravity is:
W_gravity = (m * g) * 10 * cos(30)
= m * 9.8 * 10 * cos(30)
Now, let's compare this to the overall change in gravitational potential energy.
The overall change in gravitational potential energy can be calculated as the difference between the gravitational potential energy at the end and the beginning of the motion:
ΔPE = PE_end - PE_beginning
= 98 m - 0
= 98 m
From the calculation of the work done by gravity, we can see that it is equal to the overall change in gravitational potential energy:
W_gravity = 98 m
This indicates that the work done by gravity is equal to the change in potential energy. This is expected because energy is conserved in a frictionless system.
For the second question:
The potential energy is only stored in a system when there is an external force acting against the force of gravity. This can happen when an object is lifted or raised above a certain height.
In the given scenario, the 5 kg box is lifted by a force at a constant speed of 3 meters.
To calculate the gravitational potential energy stored in the box, we can use the formula:
Gravitational potential energy (Ug) = mass (m) * acceleration due to gravity (g) * height (h)
Given that the height is 3 meters and the mass is 5 kg, we can calculate the gravitational potential energy as follows:
Ug = m * g * h
= 5 kg * 9.8 m/s^2 * 3 m
= 147 J
So, the gravitational potential energy stored in the box is 147 J.
To calculate the work done by the force to lift the box at a constant speed, we can use the formula:
Work (W) = force (F) * displacement (d) * cos(theta)
Since the box is lifted at a constant speed, the force applied must be equal in magnitude and opposite in direction to the force of gravity. Therefore, the force required to lift the box is given by:
Force = mass (m) * acceleration due to gravity (g)
= 5 kg * 9.8 m/s^2
The displacement is given as 3 meters, and the angle between the force and displacement is 0 degrees (cos(0) = 1).
Therefore, the work done by the force to lift the box is:
W = (5 kg * 9.8 m/s^2) * 3 m * cos(0)
= 147 J
We can observe that the work done by the force to lift the box is equal to the gravitational potential energy stored in the box (147 J). This indicates that the work done is equal to the energy stored, confirming the conservation of energy.
If more work is done lifting the box than what is stored as gravitational potential energy, it means that some additional energy has been either inputted into the system or dissipated as heat or other forms of energy. This would violate the conservation of energy principle.
1. To calculate the gravitational potential energy at the beginning and end of the motion, you need to use the formula:
Gravitational Potential Energy (Ug) = m * g * h
Where:
- m is the mass of the object (in this case, the box)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- h is the height or vertical distance above a reference point
At the beginning of the motion, the box is at a higher vertical distance (h1) than at the end of the motion (h2), since it slides down the incline. So, to calculate the gravitational potential energy at each point:
Ug1 = m * g * h1
Ug2 = m * g * h2
2. The work done by gravity as the box slides down the incline can be calculated using the formula:
Work (W) = m * g * d * cos(theta)
Where:
- m is the mass of the object
- g is the acceleration due to gravity
- d is the displacement along the incline (distance traveled)
- theta is the angle of the incline
To compare this to the overall change in gravitational potential energy:
Change in gravitational potential energy (ΔUg) = Ug2 - Ug1
Comparing the work done by gravity to the overall change in gravitational potential energy can give insights into the energy conservation in the system.
3. Potential Energy is only stored in a system when work is done on the system against a conservative force, such as gravity, elastic force, or electric force. These forces are called conservative because the work done by them does not depend on the path taken but only on the initial and final positions.
Therefore, the potential energy is stored in a system when an external force (like gravity) is applied to displace an object vertically, and the force is acting against gravity.
In the case of lifting a box, the work done by the force to lift the box at a constant speed can be calculated using the formula:
Work (W) = Force * Distance
To compare this answer to the amount of gravitational potential energy stored in the box:
Gravitational Potential Energy (Ug) = m * g * h
Comparing the work done by the force to the gravitational potential energy can provide insights into the energy transfer and conservation in the lifting process.
4. If you did more work lifting the box than what was stored as gravitational potential energy, it means that some of the work done is dissipated as other forms of energy, such as heat or sound. The additional work done beyond the gravitational potential energy is not converted entirely into potential energy but is lost as non-conservative work.
In practical situations, it is difficult to achieve 100% efficiency, and some energy is always lost as dissipated or wasted energy due to factors such as friction or air resistance. Therefore, it is important to consider the efficiency and energy losses in real-world scenarios to make accurate conclusions about the energy conservation and utilization in a system.