9. Work & Energy

Estimate the work you do to mow a lawn 10 m by 20 m with a 50-cm-wide mower. Assume you push with a horizontal force of about 15 N.

A 25 kg air compressor is dragged up a rough incline from ${\vec{r}}_1=(1.3\hat{ı}+1.3\hat{ȷ})\text{m}$ to ${\vec{r}}_2=(8.3\hat{ı}+2.9\hat{ȷ})\text{m}$, to where the y-axis is vertical. How much work does gravity do on the compressor during this displacement?

The energy used to pump liquids and gases through pipes is a significant fraction of the total energy consumption in the United States. Consider a small volume V of a liquid that has density ρ. Assume that the fluid is nonviscous so that friction with the pipe walls can be neglected. An upward-pushing force from a pump lifts this volume of fluid a height h at constant speed. How much work does the pump do?

A 1000 kg elevator accelerates upward at 1.0 m/s² for 10 m, starting from rest. How much work does the tension in the elevator cable do on the elevator?

The three ropes shown in the bird's-eye view of FIGURE EX9.18 are used to drag a crate 3.0 m across the floor. How much work is done by each of the three forces?

Objects that rotate in air or water experience a torque due to drag. With quadratic drag, a drag torque that's negligible at low rpm quickly becomes significant as the rpm increases. Consider a square bar with cross section a x a and length L. It is rotating on an axle through its center at angular velocity ω in a fluid of density ρ. Assume that the drag coefficient C_𝒹 is constant along the length of the bar. Find an expression for the magnitude of the drag torque on the bar. Hint: Begin by considering the drag force on a small piece of the bar of length dr at distance r from the axle.

Consider a force F(𝓍) = A𝓍^3/2 acting on an object moving in a straight line. Assume that A = 10.0 N/m^z. Calculate the work done by this force as the object moves from 𝓍 = 0 to 𝓍 = 3.0 m.

In a certain library the first shelf is 15.0 cm off the ground, and the remaining four shelves are each spaced 38.0 cm above the previous one. If the average book has a mass of 1.25 kg with a height of 22.0 cm, and an average shelf holds 28 books (standing vertically), how much work is required to fill all the shelves, assuming the books are all laying flat on the floor to start?

(II) A lever such as that shown in Fig. 7–20 can be used to lift objects we might not otherwise be able to lift. Show that the ratio of output force, F_O, to input force, F_I, is related to the lengths ℓ_I and ℓ_O from the pivot by F_O / F_I = ℓ_I / ℓ_O. Ignore friction and the mass of the lever, and assume the work output equals the work input.

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The head of a hammer with a mass of 1.2 kg is allowed to fall onto a nail from a height of 0.65 m. What is the maximum amount of work it could do on the nail? Why do people not just “let it fall” but add their own force to the hammer as it falls?

A 2800-kg space vehicle, initially at rest, falls vertically from a height of 2900 km above the Earth’s surface. Determine how much work is done by the force of gravity in bringing the vehicle to the Earth’s surface.

A package of mass m is placed onto a horizontal conveyor belt moving at speed v (Fig. 7–32). The coefficient of kinetic friction between package and belt is μ_k. How much of this work is done against friction and how much to accelerate the package?

A softball having a mass of 0.25 kg is pitched horizontally at 120 km/h. By the time it reaches the plate, it may have slowed by 10%. Neglecting gravity, estimate the average force of air resistance during a pitch, if the distance between the plate and the pitcher is about 15 m.

A constant force $\vec{F}=(2.0\hat{i}+4.0\hat{j})\text{ N}$ acts on an object as it moves along a straight-line path. If the object’s displacement is $\vec{d}=(1.0\hat{i}+5.0\hat{j})\text{ m}$, calculate the work done by using these alternate ways of writing the dot product:

(a) $W=Fd\cos \theta$; (b) $W=F_x{d}_x+F_y{d}_y$.

A child is pulling a wagon down the sidewalk. For 5.0 m the wagon stays on the sidewalk and the child pulls with a horizontal force of 22 N. Then one wheel of the wagon goes off onto the grass so the child has to pull with a horizontal force of 38 N at an angle of 12° to the side for the next 3.0 m. Finally the wagon gets back on the sidewalk so the child makes the rest of the trip, 8.5 m, with a force of 22 N. How much total work did the child do on the wagon?