10. Conservation of Energy

You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 10-m-high hill, then descends 15 m to the track's lowest point. You've determined that the spring can be compressed a maximum of 2.0 m and that a loaded car will have a maximum mass of 400 kg. For safety reasons, the spring constant should be 10% larger than the minimum needed for the car to just make it over the top. What is the maximum speed of a 350 kg car if the spring is compressed the full amount?

A horizontal spring with spring constant 100 N/m is compressed 20 cm and used to launch a 2.5 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15. Use work and energy to find how far the box slides across the rough surface before stopping.

A freight company uses a compressed spring to shoot 2.0 kg packages up a 1.0-m-high frictionless ramp into a truck, as FIGURE P10.52 shows. The spring constant is 500 N/m and the spring is compressed 30 cm. What is the speed of the package when it reaches the truck?

The elastic energy stored in your tendons can contribute up to 35% of your energy needs when running. Sports scientists find that (on average) the knee extensor tendons in sprinters stretch 41 mm while those of nonathletes stretch only 33 mm. The spring constant of the tendon is the same for both groups, 33 N/mm. What is the difference in maximum stored energy between the sprinters and the nonathletes?

In a physics lab experiment, a compressed spring launches a 20 g metal ball at a 30° angle. Compressing the spring 20 cm causes the ball to hit the floor 1.5 m below the point at which it leaves the spring after traveling 5.0 m horizontally. What is the spring constant?

A vertical spring (ignore its mass), whose spring constant is 875 N/m, is attached to a table and is compressed down by 0.220 m. What upward speed can it give to a 0.380-kg ball when released?

In which of the following scenarios is $\mathrm{elastic} \mathrm{potential} \mathrm{energy}$ present?

A spring has a spring constant k of 78.0 N/m. How much must this spring be compressed to store 45.0 J of potential energy?

(III) An engineer is designing a spring to be placed at the bottom of an elevator shaft. If the elevator cable breaks when the elevator is at a height h above the top of the spring, calculate the value that the spring constant k should have so that passengers undergo an acceleration of no more than 4.5 g when being brought to rest. Let M be the total mass of the elevator and passengers.

Which of the following scenarios generates the most elastic potential energy in a spring with spring constant $k$ and equilibrium length, assuming the same spring is used in each case?

A spring of negligible mass has force constant $k = 1600$ N/m. How far must the spring be compressed for $3.20$ J of potential energy to be stored in it?

For a spring obeying Hooke's law, doubling which of the following quantities would cause the largest increase in the spring's elastic potential energy $U$?

When a spring is stretched by an external force and held at a fixed length, does the spring possess elastic potential energy?

A 10 kg box slides 4.0 m down the frictionless ramp shown in FIGURE CP10.73, then collides with a spring whose spring constant is 250 N/m. What is the maximum compression of the spring?

A 4-kg block moving on a flat surface strikes a massless, horizontal spring of force constant 600 N/m with a 20 m/s. The block-surface coefficient of friction is 0.5. Calculate the maximum compression that the spring will experience.