A person of mass 84kg is at rest at the bottom of an inclined plane that makes an angle of 30 degrees with the horizontal. he walks up the inclined plane through a certain distance and has a speed of 0.40 m/s. How much net work has been accomplished in walking through that distance?
To calculate the net work done in walking up the inclined plane, we need to first determine the change in potential energy and kinetic energy. The net work done will be equal to the change in total mechanical energy, which is the sum of potential energy and kinetic energy.
1. Calculate the change in potential energy:
The potential energy (PE) at the bottom of the inclined plane can be calculated using the formula:
PE = m * g * h
where m is the mass (84 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height difference between the bottom and the final position on the inclined plane.
Since the inclined plane makes an angle of 30 degrees with the horizontal, we can calculate the height difference with the formula:
h = d * sin(theta)
where d is the distance covered on the inclined plane and theta is the angle of inclination (30 degrees).
2. Calculate the change in kinetic energy:
The change in kinetic energy (ΔKE) is the difference between the final and initial kinetic energy. Initially, the person is at rest, so the initial kinetic energy is zero. The final kinetic energy can be calculated using the formula:
KE = (1/2) * m * v²
where m is the mass (84 kg) and v is the final speed (0.40 m/s).
3. Calculate the net work done:
The net work (W) can be calculated using the formula:
W = ΔPE + ΔKE
where ΔPE is the change in potential energy and ΔKE is the change in kinetic energy.
Plug in the calculated values from steps 1 and 2 to find the net work done.
Here's a step-by-step breakdown of the calculation:
1. Calculate the height difference:
h = d * sin(theta)
= d * sin(30 degrees)
= d * 0.5
2. Calculate the change in potential energy:
ΔPE = m * g * h
= 84 kg * 9.8 m/s² * (d * 0.5)
3. Calculate the change in kinetic energy:
KE = (1/2) * m * v²
= (1/2) * 84 kg * (0.40 m/s)²
4. Calculate the net work done:
W = ΔPE + ΔKE
Plug in the values obtained from steps 2 and 3, and solve for the net work done.