A crate of fruit with a mass of 37.5 kg and a specific heat capacity of 3800 J/(kg⋅K) slides 7.60 m down a ramp inclined at an angle of 38.5 degrees below the horizontal.
If the crate was at rest at the top of the incline and has a speed of 2.35 m/s at the bottom, how much work Wf was done on the crate by friction? Use 9.81 m/s^2 for the acceleration due to gravity and express your answer in joules.
Wow, you sure know how to make a crate of fruit slide down a ramp sound exciting! Moving on to your question: the work done by friction can be determined using the work-energy principle, which states that the work done is equal to the change in kinetic energy.
So, let's calculate the change in kinetic energy of the crate. The initial kinetic energy (Ki) can be calculated using the equation:
Ki = (1/2) * mass * (initial velocity)^2
Ki = (1/2) * 37.5 kg * (0 m/s)^2
But since the crate is at rest at the top, its initial kinetic energy is zero. So, Ki = 0.
The final kinetic energy (Kf) can be calculated using the equation:
Kf = (1/2) * mass * (final velocity)^2
Kf = (1/2) * 37.5 kg * (2.35 m/s)^2
Now, we can calculate the change in kinetic energy (ΔK) as:
ΔK = Kf - Ki
ΔK = (1/2) * 37.5 kg * (2.35 m/s)^2 - 0
ΔK = (1/2) * 37.5 kg * (5.5025 m^2/s^2)
ΔK = 102.565625 J
So, the work done on the crate by friction (Wf) is equal to the change in kinetic energy:
Wf = ΔK = 102.565625 J
And that's the "fruitful" answer you were looking for!
To find the work done on the crate by friction, we need to consider the change in kinetic energy.
The work-energy principle states that the work done on an object equals the change in its kinetic energy. Mathematically, this can be expressed as:
Wf = ΔKE
The kinetic energy change (ΔKE) can be calculated by subtracting the initial kinetic energy (KEi) from the final kinetic energy (KEf):
ΔKE = KEf - KEi
The initial kinetic energy is given by:
KEi = (1/2) * m * v^2
where:
m = mass of the crate = 37.5 kg
v = speed at the bottom = 2.35 m/s
KEi = (1/2) * 37.5 kg * (2.35 m/s)^2
Next, we need to calculate the final kinetic energy. At the bottom of the ramp, the crate has a speed of 2.35 m/s. The kinetic energy equation remains the same:
KEf = (1/2) * m * v^2
But in this case, the acceleration due to gravity adds potential energy, which is converted into kinetic energy:
KEf = (1/2) * m * v^2 + mgh
where:
g = acceleration due to gravity = 9.81 m/s^2
h = vertical distance traveled down the ramp
To find h, we can use the given angle of inclination (θ) and the horizontal distance traveled (d).
h = d * sin(θ)
where:
θ = 38.5 degrees
d = 7.60 m
h = 7.60 m * sin(38.5 degrees)
Finally, we can substitute the values into the equation for KEf:
KEf = (1/2) * 37.5 kg * (2.35 m/s)^2 + (37.5 kg * 9.81 m/s^2 * 7.60 m * sin(38.5 degrees))
Calculating the final and initial kinetic energy, we can determine the work done by friction:
Wf = ΔKE = KEf - KEi
To find the work done on the crate by friction, we need to determine the change in kinetic energy as the crate slides down the ramp.
First, let's find the initial kinetic energy (Ki) of the crate at the top of the incline. The formula for kinetic energy is:
Ki = (1/2) * mass * velocity^2
Given:
Mass (m) = 37.5 kg
Velocity (vi) = 0 m/s (since the crate is at rest)
Ki = (1/2) * 37.5 kg * (0 m/s)^2 = 0 joules
Next, let's find the final kinetic energy (Kf) of the crate at the bottom of the incline. The formula for kinetic energy is the same:
Kf = (1/2) * mass * velocity^2
Given:
Velocity (vf) = 2.35 m/s
Kf = (1/2) * 37.5 kg * (2.35 m/s)^2 = 65.77 joules
Now, we can calculate the work done by friction (Wf) using the formula:
Work (Wf) = Kf - Ki
Wf = 65.77 joules - 0 joules = 65.77 joules
Therefore, the amount of work done on the crate by friction is 65.77 joules.
1904 J
initial PE-finalKE=workdonebyfriction
=mg*7.60*sin38.5- 1/2 m 2.35^2