Bill is moving a crate of mass 35 kg up a ramp that forms a 25 degree angle with the horizontal to a height of 5 m.
A) If the ramp is frictionless, how much work must Bill do in pushing the crate to the top?
B) How much work does gravity do as Bill pushes the crate up the ramp?
To find the work done in each scenario, we need to understand the formulas involved in calculating work.
Work (W) is defined as the force (F) applied to an object times the distance (d) over which the force is applied, and the angle (θ) between the force vector and the displacement vector. Mathematically, it can be expressed as:
W = F * d * cos(θ)
Now, let's solve the problems step by step.
A) If the ramp is frictionless, the only force Bill needs to overcome is the component of gravity parallel to the ramp's surface. This force is called the gravitational force component along the ramp.
To find the work Bill does in pushing the crate to the top, we can use the formula mentioned above. The force applied by Bill is equal and opposite to the gravitational force component along the ramp. So, we need to calculate the magnitude of this force.
1. Start by finding the gravitational force (Fg) acting on the crate. The formula for gravitational force is given by:
Fg = m * g
where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Calculate the gravitational force component along the ramp (Fg_parallel) using:
Fg_parallel = Fg * sin(θ)
In this case, θ is the angle of the ramp, which is 25 degrees.
3. Now, calculate the work done by Bill using the formula:
W = Fg_parallel * d
where d is the distance moved vertically, which is 5 meters.
Substitute the known values and solve for W to find the amount of work Bill must do to push the crate to the top.
B) Gravity also does work on the crate as Bill pushes it up the ramp. To find the work done by gravity, we use the formula from earlier and determine the force of gravity acting vertically downward:
1. Calculate the gravitational force acting vertically downward (Fg_vertical) using:
Fg_vertical = Fg * cos(θ)
2. Calculate the work done by gravity (W_gravity) using the formula:
W_gravity = Fg_vertical * d
Here, d is again the vertical distance moved, which is 5 meters.
Substitute the known values and solve for W_gravity to find the amount of work done by gravity as Bill pushes the crate up the ramp.
By following these steps, you can determine the work done by Bill and the work done by gravity in pushing the crate up the ramp.
m g h = 35 * 9.81 * 5 Joules
part B is negative part A