A mover pushes a 4.21 kg box with a 36.6 N constant horizontal force up a 10.0° ramp that has a height of 1.57 m. If the ramp is assumed to be frictionless, find the speed of the box as it reaches the top of the ramp using work and energy.
work done= force*distance where force is vertical force
force=36.6-mg*sin10 parallel to plane
distance=1.57 along plane
force*distance=finalKE+finalPE
final KE=1/2 mv^2
final PE=1.57*mg
To find the speed of the box as it reaches the top of the ramp using work and energy, we need to consider the work-energy principle. According to this principle, the work done on an object is equal to the change in its kinetic energy.
Let's break down the problem step by step:
1. Determine the work done by the mover on the box: Work is defined as the product of the force applied and the distance over which it is applied. In this case, the mover is applying a constant horizontal force, so the work done can be calculated as:
Work = Force × Distance × cos(θ)
Where:
Force = 36.6 N (given)
Distance = height of the ramp = 1.57 m (given)
θ = angle of the ramp = 10.0° (given)
Plugging in the values, we get: Work = 36.6 N × 1.57 m × cos(10.0°)
2. Determine the change in potential energy: As the box is raised to the top of the ramp, its potential energy changes. The change in potential energy can be calculated as:
Change in Potential Energy = m × g × h
Where:
m = mass of the box = 4.21 kg (given)
g = acceleration due to gravity = 9.8 m/s² (constant)
h = height of the ramp = 1.57 m (given)
Plugging in the values, we get: Change in Potential Energy = 4.21 kg × 9.8 m/s² × 1.57 m
3. Equate the work done by the mover to the change in potential energy: According to the work-energy principle, the work done is equal to the change in potential energy. So we can write:
Work = Change in Potential Energy
Equating the values calculated in steps 1 and 2, we get:
36.6 N × 1.57 m × cos(10.0°) = 4.21 kg × 9.8 m/s² × 1.57 m
4. Solve the equation for the speed of the box: Rearrange the equation to solve for the speed:
Speed = √(2 × (Change in Potential Energy / mass))
Plugging in the values, we get:
Speed = √(2 × (36.6 N × 1.57 m × cos(10.0°) / 4.21 kg × 9.8 m/s²))
Now you can calculate the speed using the given values and solve the equation.