Chapter 10: Interactions and Potential Energy

Sam, whose mass is $75kg$, straps on his skis and starts down a $50$-m-high, ${20}^o$frictionless slope. A strong headwind exerts a horizontal force of $200$N on him as he skies. Use work and energy to find Sam’s speed at the bottom.

A horizontal spring with a spring constant $100N/m$is compressed $20cm$and used to launch a $2.5kg$ 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. With what minimum speed must you toss a $100g$ball straight up to just touch the $10-m$-high roof of the gymnasium if you release the ball $1.5m$above the ground? Solve this problem using energy.

b. With what speed does the ball hit the ground?

Rank in order, from most to least, the elastic potential energy ${\left(U_{s_p}\right)}_{a}to{\left(U_{S_p}\right)}_d$stored in the springs of FIGURE $Q10.5$Explain.

Truck brakes can fail if they get too hot. In some mountainous areas, ramps of loose gravel are constructed to stop runaway trucks that have lost their brakes. The combination of a slight upward slope and a large coefficient of rolling resistance as the truck tires sink into the gravel brings the truck safely to a halt. Suppose a gravel ramp slopes upward at $6.0^o$and the coefficient of rolling friction is $0.40$. Use work and energy to find the length of a ramp that will stop a $15,000kg$ truck that enters the ramp at $35m/s\left(\approx 75mph\right)$.

A freight company uses a compressed spring to shoot $2.0kg$packages up a -high frictionless ramp into a truck, as FIGURE $P10.51$shows. The spring constant is $500N/m$and the spring is compressed $30cm$.

a. What is the speed of the package when it reaches the truck?

b. A careless worker spills his soda on the ramp. This creates a $50-cm$-long sticky spot with a coefficient of kinetic friction $0.30$. Will the next package make it into the truck?

Use work and energy to find an expression for the speed of the block in FIGURE $P10.52$ just before it hits the floor if

(a) the coefficient of kinetic friction for the block on the table is ${\mu }_k$and

(b) the table is friction less

a. A $50g$ice cube can slide without friction up and down a ${30}^o$slope. The ice cube is pressed against a spring at the bottom of the slope, compressing the spring $10cm$. The spring constant is $25N/m$. When the ice cube is released, what total distance will it travel up the slope before reversing direction?

b. The ice cube is replaced by a $50g$plastic cube whose coefficient of kinetic friction is $0.20$. How far will the plastic cube travel up the slope? Use work and energy.

The spring shown in FIGURE P10.54 is compressed $50cm$and used to launch a $100kg$physics student. The track is frictionless until it starts up the incline. The student’s coefficient of kinetic friction on the ${30}^o$incline is $0.15$.

a. What is the student’s speed just after losing contact with the spring?

b. How far up the incline does the student go?

Protons and neutrons (together called nucleons) are held together in the nucleus of an atom by a force called the strong force. At very small separations, the strong force between two nucleons is larger than the repulsive electrical force between two protons—hence its name. But the strong force quickly weakens as the distance between the protons increases. A well-established model for the potential energy of two nucleons interacting via the strong force is

$U=U_0\left[1-e^{-x/x_0}\right]$

where x is the distance between the centers of the two nucleons, x0 is a constant having the value $x_o=2.0\times {10}^{-15}m$, and $U_o=6.0\times {10}^{-11}J.$

Quantum effects are essential for a proper understanding of nucleons, but let us innocently consider two neutrons as if they were small, hard, electrically neutral spheres of mass $1.67\times {10}^{-27}kg$and diameter $1.0\times {10}^{-15}m$. Suppose you hold two neutrons $5.0\times {10}^{-15}m$apart, measured between their centers, then release them. What is the speed of each neutron as they crash together? Keep in mind that both neutrons are moving.