MMP: Ch. 10

An ice cube of mass 50.0 g can slide without friction up and down a 25.0 degree slope. The ice cube is pressed against a spring at the bottom of the slope, compressing the spring 0.100 m . The spring constant is 25.0 N/m . When the ice cube is released, how far will it travel up the slope before reversing direction? Considering only the initial and final situations, identify which mechanical energies have changed. In addition, determine whether you will need to consider any work done by external forces or changes in thermal energy. Identify the initial and final gravitational potential energies. Identify the initial and final elastic potential energies. An ice cube of mass 50.0 g can slide without friction up and down a 25.0 degree slope. The ice cube is pressed against a spring at the bottom of the slope, compressing the spring 0.100 m . The spring constant is 25.0 N/m . When the ice cube is released, how far will it travel up the slope before reversing direction? Notice that all the initial spring potential energy was transformed into gravitational potential energy. If you compressed the spring to a distance of 0.200 m , how far up the slope will an identical ice cube travel before reversing directions? Note that the spring is compressed twice as much as in the original problem.

- Gravitational potential energy has changed. - Spring potential energy has changed. (Ug)i, (Ug)f = 0. mgdsin(θ) (Us)i, (Us)f= 1/2kx^2, 0 d = 0.604 m 2.41 m

The board can only bend so much, and therefore it can only store so much elastic potential energy before it breaks. This means that _____________.

- In order to break the board, the ball must be dropped from a height greater than some minimum distance. - In order to break the board, the ball must be moving at a speed greater than some minimum speed. - The ball loses some, but not all, of its kinetic energy. It continues to move after the board breaks.

When a flea (mm = 450 μg) is jumping up, it extends its legs 0.5 mm and reaches a speed of 1 m/s in that time. How high can this flea jump? Ignore air drag and use gg = 10 m/s^2

0.050 m

Pilobolus is a genus of fungi commonly found on dung and known for launching its spores a large distance for a sporangiophore only 1 cmcm tall. To achieve this, Pilobolus accelerates its spores (mm = 10−8−8 kg) to 7.0 m/s in 2.0 μsμs. For calculations, use gg = 10 m/s2m/s2.Source: Yafetto L, Carroll L, Cui Y, Davis DJ, Fischer MWF, et al. 2008 The Fastest Flights in Nature: High-Speed Spore Discharge Mechanisms among Fungi. PLoS ONE 3(9) If the spores are shot out horizontally at the maximum speed, how far away from the fungi do they land? Neglect air resistance. Immediately after being launched horizontally to the right, in what direction does the acceleration vector point? If shot vertically, how high above the Pilobolus head would the spores go? What is the explosive force on the spores when launched? How much work is done by the Pilobolus head when it ejects the spore? If the spore is shot directly upward, how much work is done by gravity before it hits the ground below?

0.31 m Downward 2.5 m 3.5×10^−2N 2.5×10^−7J 1.0×10^−9J

A car is traveling at 11 m/s . How fast would the car need to go to double its kinetic energy? By what factor does the car's kinetic energy increase if its speed is doubled to 22 m/s ?

16 m/s K2/K1 = 4.0

A fireman of mass 80 kg slides down a pole. When he reaches the bottom, 4.2 m below his starting point, his speed is 2.2 m/s . By how much has thermal energy increased during his slide?

E_thermal = 3100 J

If the springy part of the fern were attached to one seed and the stem, then when the --------- energy of a springy part is converted to ------- energy, some of the ----------energy would be on the stem. In fact, if the stem has very little mass, then it would move a greater distance than the seed and according to -------------, more work would be done on the ------------- than on the ------------------. If there are two seeds, then each will leave with the same ---------------- energy and no energy is wasted on unwanted motion of the stem.

Elastic Potential Kinetic Kinetic W = FΔx Stem Seed Kinetic

Some runners train with parachutes that trail behind them to provide a large drag force. These parachutes are designed to have a large drag coefficient. One model expands to a square 1.8 mm on a side, with a drag coefficient of 1.4. A runner completes a 180 mm run at 6.0 m/s with this chute trailing behind. How much thermal energy is added to the air by the drag force?

Eth = 1.8×10^4 J

A 31 kg child slides down a playground slide at a constant speed. The slide has a height of 3.8 m and is 7.8 m long. Using the law of conservation of energy, find the magnitude of the kinetic friction force acting on the child.

Ff = 150 N

By the time the blocks reach the ground, they have transformed identical amounts ------- of energy into ------- energy of the blocks and energy -------- of the cylinders. But the moment of inertia of --------- a cylinder is higher than that of a ------------cylinder of the same mass, so more of the energy of the system is in the form of rotational kinetic energy for the --------------cylinder than for the one. This leaves less energy in the form of translational kinetic energy for the -----------cylinder. But it is the ---------- energy that determines the speed of the block. So the block moves more slowly for the system with the ---------cylinder, and so its block reaches the ground last.

Gravitational Potential Translational Kinetic Rotational Kinetic Hollow Solid Hollow Solid Hollow Translational Kinetic Hollow

What is the sequence of energy transformations from the moment the ball is dropped to the moment the board is bent to its maximum extent and the ball is at rest? (Ignore any losses to thermal energy.)

Gravitational potential energy to kinetic energy to elastic potential energy

When running, your body has -------- energy. When you fall, the friction between the carpet and your body transforms your initial ---------- energy into ------------- energy. Because the surfaces in contact (your knees and the carpet) are heating up, it can result in a burn.

Kinetic Kinetic Thermal

A 500 kg horse can provide a steady output power of 750 W (that is, 1 horsepower) when pulling a load. How about a 38 kg sled dog? Data show that a 38 kg dog can pull a sled that requires a pulling force of 70 N at a steady 2.4 m/s. What steady output power can the 38 kg dog provide? What is the specific power value for the dog? What is the specific power value for the horse? What is the minimum number of dogs needed to provide about the same power as one horse?

P_dog = 170 W S_Pdog = 4.4 W/kg SP_horse = 1.50 W/kg N = 5 dogs

A 710 kg car drives at a constant speed of 23 m/s . It is subject to a drag force of 500 N. What power is required from the car's engine to drive the car on level ground? What power is required from the car's engine to drive the car up a hill with a slope of 2.0 ∘∘?

P_ground = 1.15×10^4 W P_upahill = 1.7×10^4 W

Swordfish are capable of stunning output power for short bursts. A 650 kg swordfish has a cross-section area of 0.92 m^2 and a drag coefficient of 0.0091--exceptionally low due to a number of adaptations. Such a fish can sustain a speed of 30 m/s for a few seconds. (Figure 1) Assume seawater has a density of 1026 kg/m^3. What is the specific power for motion at this high speed?

SP = 180 W/kg

is the potential-energy diagram for a 35g particle that is released from rest at x = 1.0 m. Will the particle move to the right (in the positive x-direction) or to the left (in the negative x-direction), and why? What is the particle's maximum speed? At what position does the particle have maximum speed? Where are the turning points of the motion?

The particle will move to the right. Moving to the left requires the particle to have negative kinetic energy, which is impossible. vmax = 13 m/s 4 m xL, xR = 1,6 m

The balloon rebounds to a height that is less than the height from which it was dropped. The "lost" energy appears in what form?

Thermal energy

Mosses don't spread by dispersing seeds; they disperse tiny spores. The spores are so small that they will stay aloft and move with the wind, but getting them to be windborne requires the moss to shoot the spores upward. Some species do this by using a spore-containing capsule that dries out and shrinks. The pressure of the air trapped inside the capsule increases. At a certain point, the capsule pops, and a stream of spores is ejected upward at 3.6 m/s, reaching an ultimate height of 20 cm. What fraction of the initial kinetic energy is converted to the final potential energy? What happens to the "lost" energy?

Uf/Ki = 0.30 It has been transformed into thermal energy of the spores and surrounding air.

Sam's job at the amusement park is to slow down and bring to a stop the boats in the log ride. If a boat and its riders have a mass of 800 kg and the boat drifts in at 1.8 m/s how much work does Sam do to stop it?

W = -1300 J

You are pulling a child in a wagon. The rope handle is inclined upward at a 60∘∘ angle. The tension in the handle is 20 N. How much work do you do if you pull the wagon 150 m at a constant speed?

W = 1.5 kJ

A 1.6 kg book is lying on a 0.80 m -high table. You pick it up and place it on a bookshelf 2.0 m above the floor. During this process, how much work does gravity do on the book? During this process, how much work does your hand do on the book?

Wg = -19 J Wh = 19 J

When the balloon hits the ground, it rebounds slightly. What is the source of the energy for this rebound?

When the balloon hits the ground, the rubber envelope stretches, storing elastic potential energy; this elastic potential energy is converted to the gravitational potential energy of the rebounding balloon.

A box of weight w=2.0N accelerates down a rough plane that is inclined at an angle ϕ=30∘ above the horizontal, as shown (Figure 6). The normal force acting on the box has a magnitude n=1.7N, the coefficient of kinetic friction between the box and the plane is μk=0.30, and the displacement d of the box is 1.8 m down the inclined plane. What is the work Ww done on the box by the weight of the box? What is the work Wn done on the box by the normal force? What is the work Wfk done on the box by the force of kinetic friction?

Ww = 1.8J 0 J -0.92 J

A 1500 kg car is approaching the hill shown in (Figure 1) at 12 m/s when it suddenly runs out of gas. Neglect any friction. Can the car make it to the top of the hill by coasting? What is the car's speed after coasting down the other side?

Yes, the car will pass the top of the hill and will continue to move down. 16 m/s

The board can only bend so much, and therefore can only store so much elastic potential energy, before it breaks. This means that, in order to break the board with your hand, _____________.

Your hand must be moving faster than some minimum speed.

An air-track glider undergoes a perfectly inelastic collision with an identical glider that is initially at rest. What fraction of the first glider's initial kinetic energy is transformed into thermal energy in this collision?

f = 0.50

The maximum energy a bone can absorb without breaking is surprisingly small. For a healthy human of mass 58 kg , experimental data show that the leg bones can absorb about 200 J . From what maximum height could a person jump and land rigidly upright on both feet without breaking his legs? Assume that all the energy is absorbed in the leg bones in a rigid landing. Neglect any friction.

h = 0.35m If a person flexes her legs on landing, the energy is transformed not only into elastic energy in the bones, but also into elastic energy in other tissues and into thermal energy.

The world's fastest humans can reach speeds of about 11 m/s. In order to increase his gravitational potential energy by an amount equal to his kinetic energy at full speed, how high would such a sprinter need to climb?

h = 6.2 m

As shown in the figure (Figure 1), a superball with mass mmm equal to 50 grams is dropped from a height of hi=1.5m . It collides with a table, then bounces up to a height of hf=1.0m . The duration of the collision (the time during which the superball is in contact with the table) is stc=15ms . In this problem, take the positive y direction to be upward, and use g=9.8m/s2 for the magnitude of the acceleration due to gravity. Neglect air resistance. Find the y component of the momentum, p_before, y of the ball immediately before the collision. Find the y component of the momentum of the ball immediately after the collision, that is, just as it is leaving the table. Find Jy, the y component of the impulse imparted to the ball during the collision. Find the y component of the time-averaged force F_avg, y in newtons, that the table exerts on the ball. Find K_after−K_before, the change in the kinetic energy of the ball during the collision, in joules.

p_before, y= -0.27 kg⋅m/s p_after, y = 0.22 kg⋅m/s Jy = 0.49 kg⋅m/s F_avg_y = 33 N K_after−K_before = -0.25 J

A 55 kg skateboarder wants to just make it to the upper edge of a "half-pipe" with radius r of 3.6 m What speed does he need at the bottom if he will coast all the way up? Consider the skateboarder as a simple particle. In reality the skateboarder isn't a simple particle: Assume that his mass in a deep crouch is concentrated 0.75 m from the half-pipe. If he remains in that position all the way up, what initial speed does he need to reach the upper edge?

vi = 8.4 m/s vi_crouch = 7.5 m/s

At normal temperatures and pressures, hydrogen gas is composed of H2 molecules. How far apart are the individual atoms in a molecule of H2?

x = 7.5×10^−2 nm

Mark pushes his broken car 180 m down the block to his friend's house. He has to exert a 110 N horizontal force to push the car at a constant speed. How much thermal energy is created in the tires and road during this short trip?

ΔETh = 20 kJ

A 76 kg bike racer climbs a 1200-m-long section of road that has a slope of 4.3 ∘. By how much does his gravitational potential energy change during this climb?

ΔU = 6.7×10^4 J