Exercise 6: Suppose you drop a 2 kg book to a friend who stands on the ground at distance D = 10 m below. If your friend's outstretched hands are at distance d = 1.5 m above the ground, (a) how much work Wg does the gravitational force do on the book as it drops to her hands? (b) What is the change AU in the gravitational potential energy of the book-Earth system during the drop? If the gravitational potential energy U of that system is taken to be zero at ground level, what is U (c) when the book is released and (d) when it reaches his hands?
To find the answers to the questions, we'll need to use the formulas related to work, gravitational potential energy, and distance.
(a) The work done by the gravitational force on the book can be determined using the formula: Work (Wg) = Force x Distance. The force of gravity can be calculated using the formula: Force (F) = mass x acceleration due to gravity. In this case, the mass of the book is given as 2 kg, and the acceleration due to gravity is approximately 9.8 m/s².
First, calculate the force of gravity on the book:
F = 2 kg x 9.8 m/s² = 19.6 N
Next, calculate the work done by the gravitational force:
Wg = F x Distance = 19.6 N x 10 m = 196 J
Therefore, the gravitational force does 196 Joules of work on the book.
(b) The change in gravitational potential energy (ΔU) of the book-Earth system during the drop is equal to the work done by gravity. This can be calculated using the formula: ΔU = - Wg (since work done by gravity is negative).
Therefore, ΔU = -196 J.
(c) Since the gravitational potential energy U is taken to be zero at ground level, it means that the initial potential energy of the book when it is released is zero. So, U when the book is released (U_initial) is zero.
(d) To calculate U when the book reaches your friend's hands (U_final), we can use the formula: U = mgh, where m is the mass of the book, g is the acceleration due to gravity, and h is the height.
First, calculate the height from the point of release to your friend's hands:
h = D - d = 10 m - 1.5 m = 8.5 m
Next, substitute the values into the formula to find U_final:
U_final = 2 kg x 9.8 m/s² x 8.5 m = 166.6 J
Therefore, U when the book reaches your friend's hands is 166.6 Joules.
To calculate the work done by the gravitational force on the book, we need to determine the vertical distance the book travels, which is given by d = D - d = 10 m - 1.5 m = 8.5 m.
(a) The work done by the gravitational force is calculated using the equation W = mgh, where m is the mass of the book (2 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical distance traveled by the book.
Wg = mgh = (2 kg) * (9.8 m/s^2) * (8.5 m) = 166.6 Joules
Therefore, the gravitational force does approximately 166.6 Joules of work on the book.
(b) The change in gravitational potential energy (ΔU) is given by the equation ΔU = -Wg, where Wg is the work done by the gravitational force.
ΔU = -Wg = -166.6 Joules
The negative sign indicates a decrease in gravitational potential energy.
(c) When the book is released, its gravitational potential energy is zero at ground level (U = 0).
(d) When the book reaches your friend's hands, its gravitational potential energy is equal to the work done by the gravitational force, but with the opposite sign.
U = -Wg = -166.6 Joules
Therefore, (c) the gravitational potential energy of the book-Earth system is zero when the book is released and (d) -166.6 Joules when it reaches your friend's hands.