A 48.4 kg keg of beer rolls down a 2.65 m long plank leading from the flatbed of a truck 1.180 m above the ground. Determine the amount of work (in Joules) done on the keg by gravity
To determine the amount of work done on the keg by gravity, you need to use the equation:
Work (W) = force (F) × displacement (d) × cos(θ)
Where:
- Work (W) is the amount of work done (in Joules).
- Force (F) is the force applied by gravity on the keg (in Newtons).
- Displacement (d) is the distance the keg is moved (in meters).
- θ is the angle between the force and the displacement.
In this case, the force applied by gravity on the keg is equal to its weight, which can be calculated using the formula:
Weight (W) = mass (m) × acceleration due to gravity (g)
Given:
- Mass of the keg (m) = 48.4 kg
- Acceleration due to gravity (g) = 9.81 m/s²
First, calculate the weight of the keg:
Weight (W) = 48.4 kg × 9.81 m/s² = 474.1044 N
Next, calculate the displacement (d) of the keg:
Displacement (d) = 2.65 m
Since the plank is inclined, the angle θ should be considered. From the information given, you can determine the angle using trigonometry. The height (h) of the truck above the ground can be calculated as follows:
Height (h) = 1.180 m
Using the given values, you can create a right triangle with the height (h), displacement (d), and the hypotenuse of the triangle being the distance the keg rolls down the plank.
Now, you can calculate the angle θ using the tangent function:
tan(θ) = h / d
θ = arctan(h / d)
Substituting the values, you get:
θ = arctan(1.180 m / 2.65 m)
Now, calculate the angle θ:
θ = arctan(0.44528301886792453)
θ ≈ 24.801°
Finally, substitute the values into the work equation:
Work (W) = 474.1044 N × 2.65 m × cos(24.801°)
Calculate the cos(24.801°) value and substitute it:
cos(24.801°) ≈ 0.912
Work (W) = 474.1044 N × 2.65 m × 0.912
Work (W) ≈ 1,114.51 Joules
Therefore, the amount of work done on the keg by gravity is approximately 1,114.51 Joules.