9. Work & Energy

In a hydroelectric dam, water falls 25 m and then spins a turbine to generate electricity. What is $\Delta U_G$ of 1.0 kg of water?

A cable with 20.0 N of tension pulls straight up on a 1.50 kg block that is initially at rest. What is the block's speed after being lifted 2.00 m? Solve this problem using work and energy.

A 1500 kg car traveling at 10 m/s suddenly runs out of gas while approaching the valley shown in FIGURE EX10.11. The alert driver immediately puts the car in neutral so that it will roll. What will be the car's speed as it coasts into the gas station on the other side of the valley? Ignore rolling friction.

A pendulum is made by tying a 500 g ball to a 75-cm-long string. The pendulum is pulled 30° to one side, then released. What is the ball's speed at the lowest point of its trajectory?

A 55 kg skateboarder wants to just make it to the upper edge of a 'quarter pipe,' a track that is one-quarter of a circle with a radius of 3.0 m. What speed does he need at the bottom?

A system of two objects has $\Delta K_{tot}=7J$ and $\Delta U=-5J$. How much work is done by interaction forces?

(II) A 66.5-kg hiker starts at an elevation of 1150 m and climbs to the top of a peak 2660 m high. Can the actual work done be greater than this? Explain.

A 66.5-kg hiker starts at an elevation of 1150 m and climbs to the top of a peak 2660 m high. 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 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). Assume a stretchy rope behaves as a spring with spring constant k. Applying the work-energy principle, show that $x=\frac{mg}{k}[1+\sqrt{1+\frac{4k\ell }{mg}}]$.

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: the average force exerted on him by the snow to stop him.

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). Assume a stretchy rope behaves as a spring with spring constant k. 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 the work-energy principle.

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?