A lawn mower is pushed a distance of 100 ft. along a horizontal path by a constant force of 60 lbs. The handle of the lawn mower is held at a downward angle of 45 degrees toward the horizontal. Explain what each value listed in this situation represents in relation to the formula W=F*PQ=llFll*lPQl cos theta. Then find the work done pushing the lawn mower.
Ah, the world of lawn mowing physics! Let's break down the situation and explain the values in relation to the formula.
W represents the work done, F represents the force applied, PQ represents the displacement, and θ represents the angle between the force and displacement vectors.
In this case, the constant force applied is 60 lbs. This force is the magnitude of F, and it's being applied in the horizontal direction to push the lawn mower.
The displacement, PQ, represents the distance the lawn mower was pushed, which in this case is 100 ft. It's the magnitude of the displacement vector.
Now, we have the angle θ. In this scenario, the handle of the lawn mower is held at a downward angle of 45 degrees toward the horizontal. This angle is important because it affects the angle between the force and the displacement vectors.
To calculate the work done, we use the formula W = F * PQ * cos θ. In this case, the force (F) is 60 lbs, the displacement (PQ) is 100 ft, and the angle (θ) is 45 degrees.
Now let's crunch the numbers with a little clown math magic! W = 60 lbs * 100 ft * cos 45 degrees.
And the grand result is... *drumroll*... W = 6000 lbs-ft! Hooray, work has been done!
In this situation, let's break down the components of the equation W = F * PQ = |F| * |PQ| * cos(theta).
W represents the work done by the force exerted to move an object along a path. It is measured in joules (J).
F refers to the applied force on the object. In this case, it is a constant force of 60 lbs (pounds). However, it is important to convert this force to the SI unit of newtons (N) because the formula requires forces to be in newtons.
PQ represents the displacement or distance moved by the object along a path. It is measured in feet (ft). Similarly, it is important to convert this distance to the SI unit of meters (m) because the formula requires distances to be in meters.
theta denotes the angle between the applied force vector and the displacement vector. In this case, the angle is 45 degrees (°).
First, let's convert the force and the distance into SI units:
1 lb = 0.45359237 kg (conversion factor)
60 lbs * 0.45359237 kg/lb ≈ 27.216 N
1 ft = 0.3048 m (conversion factor)
100 ft * 0.3048 m/ft = 30.48 m
Now we can calculate the work done using the formula:
W = |F| * |PQ| * cos(theta)
W = 27.216 N * 30.48 m * cos(45°)
cos(45°) is √2 / 2
W ≈ 27.216 N * 30.48 m * (√2 / 2)
W ≈ 1209.65 J
Therefore, the work done pushing the lawn mower is approximately 1209.65 joules (J).
In this situation, we have the following values:
- F represents the magnitude of the constant force applied, which is 60 lbs (pounds). It is the force with which the lawn mower is being pushed.
- PQ represents the displacement or distance over which the force is applied, which is 100 ft (feet). It is the distance the lawn mower travels.
- θ represents the angle between the direction of the force and the direction of the displacement. In this case, it is 45 degrees because the handle is held at a downward angle of 45 degrees toward the horizontal.
Now, to find the work done pushing the lawn mower, we can use the formula W = F * PQ * cos(θ).
Plugging in the given values:
W = 60 lbs * 100 ft * cos(45 degrees)
To calculate the cosine of 45 degrees, we use a scientific calculator or the trigonometric table. The cosine of 45 degrees is √2/2 ≈ 0.7071.
Substituting the value of cos(45 degrees) into the equation:
W = 60 lbs * 100 ft * 0.7071
Now, we can simplify the equation:
W = 4242.64 lbs * ft
Therefore, the work done pushing the lawn mower is approximately 4242.64 lbs * ft (pounds times feet).
work = Force * distance in direction of force
= Force vector dot Distance vector
= |F| |D| cos angle between the two vectors
= 60 * 100 * sqrt 2 / 2