A student librarian picks up a 2.2 kg book from the floor to a height of 1.05 m. He carries the book 8.3 m to the stacks and places the book on a shelf that is 0.35 m above the floor. How much work does he do on the book?
To calculate the work done by the student librarian on the book, we can use the formula:
Work = Force × Distance × Cos(θ)
In this case, the force exerted by the student librarian is equal to the weight of the book, which can be calculated using the formula:
Force = Mass × Acceleration due to gravity
The mass of the book is given as 2.2 kg, and the acceleration due to gravity is approximately 9.8 m/s².
The distance covered by the student librarian is the total vertical distance, which includes lifting the book from the floor to the shelf, and the horizontal distance of carrying the book to the shelf.
Thus, the total distance covered is equal to the sum of the vertical distance (1.05 m) and the horizontal distance (8.3 m).
To calculate the angle (θ) between the force applied and the direction of motion (in this case, the vertical direction), we can use the formula:
Cos(θ) = Adjacent / Hypotenuse
The adjacent side is the vertical distance (1.05 m) and the hypotenuse is the total distance covered (1.05 m + 8.3 m). By dividing the adjacent by the hypotenuse, we can find the cosine of the angle.
Finally, we can substitute the values into the work formula to calculate the work done on the book.
Let's calculate the work step by step:
1. Calculate the force:
Force = Mass × Acceleration due to gravity
Force = 2.2 kg × 9.8 m/s²
2. Calculate the total distance covered:
Total Distance = Vertical Distance + Horizontal Distance
Total Distance = 1.05 m + 8.3 m
3. Calculate the angle (θ) using the cosine formula:
Cos(θ) = Vertical Distance / Total Distance
θ = Cos⁻¹(Vertical Distance / Total Distance)
4. Calculate the work:
Work = Force × Total Distance × Cos(θ)
By plugging in the values we found, we can calculate the work done on the book.