9. Work & Energy

Consider a force F(𝓍) = A𝓍³⸍² acting on an object moving in a straight line. Assume that A = 10.0 N/mᶻ.

(b) Calculate the work done by this force as the object moves from 𝓍 = 0 to 𝓍 = 3.0m

(II) 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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(I) 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?

(III) 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 μₖ . <IMAGE>

(d) 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.

(II) A constant force F→ = (2.0 î + 4.0 ĵ) N acts on an object as it moves along a straight-line path. If the object’s displacement is d→ = (1.0 î + 5.0 ĵ) m, calculate the work done by using these alternate ways of writing the dot product:

(a) W = Fd cosθ ; (b) W = Fₓdₓ + Fᵧdᵧ .

(II) 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?

An airplane pilot fell 370 m after jumping from an aircraft without his parachute opening. He landed in a snowbank, creating a crater 1.1 m deep, but survived with only minor injuries. Assuming the pilot’s mass was 82 kg and his terminal velocity was 45 m/s, estimate:

(c) the work done on him by air resistance as he fell. Model him as a particle.

(II) Consider a force F₁ = A /√x which acts on an object during its journey along the x axis from x = 0.0 to x = 1.0m, where A = 3.0 N • m¹⸍² . Show that during this journey, even though F₁ is infinite at x = 0.0, the work W done on the object by this force is finite, and determine W.

(III) A 3.0-m-long steel chain is stretched out along the top level of a horizontal scaffold at a construction site, in such a way that 2.0 m of the chain remains on the top level and 1.0 m hangs vertically, Fig. 7–27. At this point, the force on the hanging segment is sufficient to pull the entire chain over the edge. Once the chain is moving, the kinetic friction is so small that it can be neglected. How much work is performed on the chain by the force of gravity as the chain falls from the point where 2.0 m remains on the scaffold to the point where the entire chain has left the scaffold? (Assume that the chain has a linear weight density of 24 N/m.) <IMAGE>

Two forces, F₁→ = (1.50î - 0.80 ĵ + 0.70k̂) N and F₂→ = ( - 0.70î + 1.20 ĵ) N, are applied on a moving object of mass 0.20 kg. The displacement vector of the object while the two forces act is d→ = (6.0 î + 8.0 ĵ + 5.0 k̂ ) m . What is the work done by the two forces?