9. Work & Energy

You pull a 3kg box on a flat surface. The coefficient of kinetic friction is 0.6. When you pull the box horizontally through a distance of 10m, it accelerates at 2m/s2. Find the net work on the box.

To pull a 51 kg crate across a rough floor, a worker applies a force of 100 N, directed 37°above the horizontal. The coefficient of friction is 0.16. If the crate moves 3.0 m, what is the total work done on the crate?

A box slides across the floor with an initial speed of 3.5m/s. If the coefficient of kinetic friction is 0.15, how far will the box slide before stopping completely?

(II) An 85-g arrow is fired from a bow whose string exerts an average force of 105 N on the arrow over a distance of 75 cm. What is the speed of the arrow as it leaves the bow?

(II) A 66.5-kg hiker starts at an elevation of 1150 m and climbs to the top of a peak 2660 m high.

(c) Can the actual work done be greater than this? Explain.

(II) A 66.5-kg hiker starts at an elevation of 1150 m and climbs to the top of a peak 2660 m high.

(b) What is the minimum work the hiker must do?

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:

(a) the work done by the snow in bringing him to rest; .

Stretchable ropes are used to safely arrest the fall of rock climbers. Suppose one end of a rope with unstretched length ℓ is anchored to a cliff and a climber of mass m is attached to the other end. When the climber is a height ℓ above the anchor point, he slips and falls under the force of gravity for a distance 2ℓ , after which the rope becomes taut and stretches a distance x as it stops the climber (see Fig. 7–37). <IMAGE>

Assume a stretchy rope behaves as a spring with spring constant k.

(a) Applying the work-energy principle, show that

x = mg/k [ 1 + √1 + (4kℓ/mg)]

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:

(b) the average force exerted on him by the snow to stop him; and

Stretchable ropes are used to safely arrest the fall of rock climbers. Suppose one end of a rope with unstretched length ℓ is anchored to a cliff and a climber of mass m is attached to the other end. When the climber is a height ℓ above the anchor point, he slips and falls under the force of gravity for a distance 2ℓ , after which the rope becomes taut and stretches a distance x as it stops the climber (see Fig. 7–37). <IMAGE>

Assume a stretchy rope behaves as a spring with spring constant k.

(b) Assuming m = 85kg , ℓ = 8.0 m and, k = 850 N/m determine x/ℓ (the fractional stretch of the rope) and kx / mg (the force that the rope exerts on the climber compared to his own weight) at the moment the climber’s fall has been stopped.

(II) How much work would be required to move a satellite of mass m from a circular orbit of radius r₁ = 2r_E about the Earth to another circular orbit of radius r₂ = 3r_E? (r_E is the radius of the Earth.)

A 25-g projectile is fired into a cube of ballistic gel at a velocity of 360 m/s. If the projectile penetrates 15 cm into the gel before stopping, find the average force exerted by the gel onto the projectile. Use

(b) the work-energy principle.

(II) A car traveling at a velocity v can stop in a minimum distance d. What would be the car’s minimum stopping distance if it were traveling at a velocity of 2v?

(II) At an accident scene on a level road, investigators measure a car’s skid mark to be 78 m long. It was a rainy day and the coefficient of friction was estimated to be 0.30.

(a) Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes.

(II) At an accident scene on a level road, investigators measure a car’s skid mark to be 78 m long. It was a rainy day and the coefficient of friction was estimated to be 0.30.

(b) Why does the car’s mass not matter?

(II) At an accident scene on a level road, investigators measure a car’s skid mark to be 78 m long. It was a rainy day and the coefficient of friction was estimated to be 0.30.

(c) What is wrong with a car that skids (see page 131)?

II) A 1.60-m-tall person lifts a 1.65-kg book off the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to

(c) How is the work done by the person related to the answers in parts (a) and (b)?

In the game of paintball, players use guns powered by pressurized gas to propel 33-g gel capsules filled with paint at the opposing team. Game rules dictate that a paintball cannot leave the barrel of a gun with a speed greater than 85 m/s. Model the shot by assuming the pressurized gas applies a constant force F to a 33-g capsule over the length of the 32-cm barrel. Determine F by

(a) using the work-energy principle, and

(II) A meteorite has a speed of 90.0 m/s when 750 km above the Earth. It is falling vertically (ignore air resistance) and strikes a bed of sand in which it is brought to rest in 3.25 m.

(b) How much work does the sand do to stop the meteorite (mass = 575 kg)?

(II) A NASA satellite is said to have observed an asteroid that is on a collision course with the Earth. The asteroid has an estimated mass, based on its size, of 4 x 10⁹ kg. It is approaching the Earth on a head-on course with a velocity of 600 m/s relative to the Earth and is now 5.0 x 10⁶ km away. With what speed will it hit the Earth’s surface, neglecting friction with the atmosphere?

We usually neglect the mass of a spring if it is small compared to the mass attached to it. But in some applications, the mass of the spring must be taken into account. Consider a spring of unstretched length ℓ and mass M_S uniformly distributed along the length of the spring. A mass m is attached to the end of the spring. One end of the spring is fixed and the mass m is allowed to vibrate horizontally without friction (Fig. 7–31). Each point on the spring moves with a velocity proportional to the distance from that point to the fixed end. For example, if the mass on the end moves with speed v₀, the midpoint of the spring moves with speed v₀ / 2. <IMAGE>

Show that the kinetic energy of the mass plus spring when the mass m is moving with velocity v is

K = (1/2)Mv²

where M = m + (1/3)M_S is the “effective mass” of the system. [Hint: Let D be the total length of the stretched spring. Then the velocity of an infinitesimal length dx of spring, of mass dM, located at x is v(x) = v₀(x/D). Note also that dM = dx( M_S/D) .]

(II) A 3.5-kg object moving in two dimensions initially has a velocity v₁→ (10.0 î + 20.0 ĵ) m/s. A net force F→ then acts on the object for 2.0 s, after which the object’s velocity is v₂→ (15.0 î + 30.0 ĵ) m/s . Determine the work done by F→ on the object.

A 2.0-kg block slides across a rough surface with a constant coefficient of kinetic friction of 0.50 (Fig. 7–38a). The block starts at x= 0 with an initial velocity of 4.9 m/s. Pushing the block is a force directed at 36.8° below the horizontal and whose magnitude increases with position as shown in Fig. 7–38b.

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(d) Draw a line on the graph showing the magnitude of the friction force versus distance x.