A mass with an initial combined potential energy and kinetic energy of 10,000 joules is dropped onto a fence post to drive it into the ground. The post is driven 0.25 meters into the ground before it comes to rest with the mass sitting on top of the post. How much work was done on the post by the mass? (The post is for a large corral for gazelles)
To find the work done on the post by the mass, we can use the work-energy principle. The principle states that the work done on an object is equal to the change in its kinetic energy.
Given:
Initial combined potential energy and kinetic energy (E_initial) = 10,000 J
Post displacement (d) = 0.25 m
Since the mass is dropped onto the post, its initial kinetic energy would become zero when it comes to rest on top of the post. The potential energy is converted into work done on the post to drive it into the ground.
Using the work-energy principle, the work done on the post (W) is equal to the change in energy:
W = E_final - E_initial
Since the final kinetic energy is zero (mass at rest), we have:
W = 0 - E_initial
Plugging in the values:
W = 0 - 10,000 J
W = -10,000 J
The negative sign indicates that work is done on the post by the mass. Therefore, the magnitude of the work done on the post by the mass is 10,000 joules.
To calculate the work done on the post by the mass, we can use the concept of work-energy theorem, which states that the work done on an object is equal to the change in its energy.
In this case, the initial energy of the mass is the combined potential and kinetic energy, which is given as 10,000 joules.
The final energy of the mass is the potential energy when it is sitting on top of the post. Since the mass is at rest, its kinetic energy is zero. Therefore, the final energy is equal to the potential energy.
We can calculate the potential energy using the formula: Potential Energy = mass * gravity * height
Given:
- The height is the distance the post is driven into the ground, which is 0.25 meters.
- The mass is not specified in the question, so we will assume it to be some value, let's say "m".
- The acceleration due to gravity is 9.8 m/s^2.
Potential Energy = m * 9.8 * 0.25
Now, to find the work done on the post, we can subtract the final energy from the initial energy:
Work done = Initial energy - Final energy
= 10,000 joules - (m * 9.8 * 0.25) joules
= 10,000 - 2.45m joules
So, the work done on the post by the mass is (10,000 - 2.45m) joules.