Chapter 1 Review Introduction to the Human Body

8 Potential Energy and Conservation of Energy

8 Chapter Review

Key Terms

conservative force

force that does work independent of path

conserved quantity

one that cannot be created or destroyed, but may exist transformed between unlike forms of itself

energy conservation

total energy of an isolated system is constant

equilibrium point

position where the assumed conservative, net strength on a particle, given by the slope of its potential energy curve, is zero

exact differential

is the full differential of a function and requires the use of fractional derivatives if the function involves more than than ane dimension

mechanical free energy

sum of the kinetic and potential energies

not-conservative force

force that does piece of work that depends on path

not-renewable

energy source that is not renewable, simply is depleted by human being consumption

potential energy

function of position, energy possessed past an object relative to the arrangement considered

potential free energy diagram

graph of a particle's potential free energy as a role of position

potential energy difference

negative of the work done acting between two points in space

renewable

free energy source that is replenished by natural processes, over homo time scales

turning point

position where the velocity of a particle, in one-dimensional motion, changes sign

Central Equations

Difference of potential energy	Δ U A B = U B − U A = − Due west A B ΔUAB=UB−UA=−WAB
Potential energy with respect to zero of potential free energy at	r ⃗ 0 Δ U = U ( r ⃗ ) − U ( r ⃗ 0 ) r→0ΔU=U(r→)−U(r→0)
Gravitational potential free energy most Earth'southward surface	U ( y ) = one thousand one thousand y + const . U(y)=mgy+const.
Potential free energy for an ideal spring	U ( x ) = one 2 k x 2 + const . U(x)=12kx2+const.
Piece of work washed by bourgeois force over a airtight path	West airtight path = ∮ Eastward ⃗ cons · d r ⃗ = 0 Wclosed path=∮East→cons·dr→=0
Status for bourgeois force in two dimensions	( d F x d y ) = ( d F y d x ) (dFxdy)=(dFydx)
Bourgeois forcefulness is the negative derivative of potential energy	F l = − d U d l Fl=−dUdl
Conservation of energy with no non-bourgeois forces	0 = Westward n c , A B = Δ ( K + U ) A B = Δ Eastward A B . 0=Wnc,AB=Δ(Grand+U)AB=ΔEAB.

Summary

8.one Potential Energy of a Arrangement

• For a single-particle organisation, the departure of potential free energy is the reverse of the work done past the forces acting on the particle as it moves from i position to some other.
• Since merely differences of potential free energy are physically meaningful, the goose egg of the potential energy role can be called at a user-friendly location.
• The potential energies for Earth'southward constant gravity, near its surface, and for a Hooke's law force are linear and quadratic functions of position, respectively.

viii.2 Conservative and Non-Conservative Forces

• A conservative force is one for which the piece of work done is independent of path. Equivalently, a forcefulness is conservative if the work washed over any closed path is cypher.
• A non-bourgeois forcefulness is 1 for which the piece of work washed depends on the path.
• For a bourgeois strength, the infinitesimal work is an exact differential. This implies conditions on the derivatives of the force'southward components.
• The component of a conservative strength, in a detail direction, equals the negative of the derivative of the potential energy for that strength, with respect to a displacement in that direction.

8.iii Conservation of Energy

• A conserved quantity is a concrete property that stays constant regardless of the path taken.
• A form of the work-energy theorem says that the modify in the mechanical energy of a particle equals the work done on information technology past not-bourgeois forces.
• If non-bourgeois forces do no piece of work and there are no external forces, the mechanical energy of a particle stays abiding. This is a argument of the conservation of mechanical energy and there is no alter in the total mechanical energy.
• For one-dimensional particle motion, in which the mechanical energy is constant and the potential energy is known, the particle's position, equally a function of time, can be constitute by evaluating an integral that is derived from the conservation of mechanical free energy.

8.four Potential Energy Diagrams and Stability

• Interpreting a one-dimensional potential energy diagram allows yous to obtain qualitative, and some quantitative, data about the movement of a particle.
• At a turning indicate, the potential energy equals the mechanical energy and the kinetic energy is zero, indicating that the management of the velocity reverses in that location.
• The negative of the slope of the potential energy curve, for a particle, equals the one-dimensional component of the conservative force on the particle. At an equilibrium point, the slope is goose egg and is a stable (unstable) equilibrium for a potential free energy minimum (maximum).

8.5 Sources of Energy

• Energy tin be transferred from one organisation to some other and transformed or converted from i type into another. Some of the bones types of energy are kinetic, potential, thermal, and electromagnetic.
• Renewable free energy sources are those that are replenished by ongoing natural processes, over human time scales. Examples are wind, water, geothermal, and solar ability.
• Non-renewable energy sources are those that are depleted by consumption, over human time scales. Examples are fossil fuel and nuclear power.

Conceptual Questions

viii.1 Potential Energy of a System

i.

The kinetic energy of a organization must always be positive or zero. Explain whether this is true for the potential energy of a arrangement.

2 .

The force exerted by a diving board is conservative, provided the internal friction is negligible. Assuming friction is negligible, describe changes in the potential free energy of a diving board as a swimmer drives from it, starting just before the swimmer steps on the board until just afterward his anxiety leave it.

3.

Describe the gravitational potential energy transfers and transformations for a javelin, starting from the bespeak at which an athlete picks up the javelin and ending when the javelin is stuck into the ground later being thrown.

4 .

A couple of soccer balls of equal mass are kicked off the basis at the aforementioned speed but at unlike angles. Soccer brawl A is kicked off at an bending slightly above the horizontal, whereas ball B is kicked slightly below the vertical. How exercise each of the following compare for ball A and brawl B? (a) The initial kinetic energy and (b) the alter in gravitational potential free energy from the ground to the highest point? If the free energy in part (a) differs from part (b), explain why there is a difference betwixt the ii energies.

5.

What is the dominant factor that affects the speed of an object that started from rest down a frictionless incline if the only work done on the object is from gravitational forces?

6 .

2 people observe a leafage falling from a tree. 1 person is standing on a ladder and the other is on the ground. If each person were to compare the energy of the leaf observed, would each person find the following to be the same or different for the leaf, from the point where it falls off the tree to when it hits the basis: (a) the kinetic free energy of the leafage; (b) the modify in gravitational potential energy; (c) the final gravitational potential energy?

8.two Conservative and Non-Conservative Forces

7.

What is the physical significant of a non-conservative force?

8 .

A bottle rocket is shot direct upward in the air with a speed 30 grand/south 30m/southward . If the air resistance is ignored, the canteen would go upwardly to a height of approximately 46 1000 46m . However, the rocket goes up to merely 35 chiliad 35m  before returning to the footing. What happened? Explain, giving only a qualitative response.

nine.

An external force acts on a particle during a trip from one point to another and back to that aforementioned point. This particle is merely effected by conservative forces. Does this particle's kinetic energy and potential energy modify as a result of this trip?

eight.3 Conservation of Energy

10 .

When a trunk slides down an inclined plane, does the work of friction depend on the body'southward initial speed? Answer the same question for a body sliding downward a curved surface.

11.

Consider the following scenario. A car for which friction isnon negligible accelerates from rest down a colina, running out of gasoline after a short distance (meet below). The commuter lets the automobile coast farther downwards the hill, then up and over a minor crest. He then coasts downward that colina into a gas station, where he brakes to a stop and fills the tank with gasoline. Place the forms of energy the car has, and how they are changed and transferred in this series of events.

12 .

A dropped brawl bounces to i-half its original meridian. Hash out the energy transformations that accept place.

xiii.

" E = G + U E=K+U  abiding is a special case of the work-free energy theorem." Discuss this argument.

14 .

In a common physics sit-in, a bowling ball is suspended from the ceiling past a rope.

The professor pulls the ball away from its equilibrium position and holds it adjacent to his nose, equally shown below. He releases the brawl and then that it swings direct abroad from him. Does he get struck by the brawl on its return swing? What is he trying to show in this demonstration?

15.

A child jumps up and down on a bed, reaching a higher acme afterwards each bounce. Explain how the child tin can increase his maximum gravitational potential free energy with each bounce.

16 .

Can a non-bourgeois force increment the mechanical energy of the system?

17.

Neglecting air resistance, how much would I accept to heighten the vertical meridian if I wanted to double the touch on speed of a falling object?

eighteen .

A box is dropped onto a spring at its equilibrium position. The spring compresses with the box fastened and comes to rest. Since the spring is in the vertical position, does the change in the gravitational potential energy of the box while the jump is compressing demand to be considered in this trouble?

Bug

8.1 Potential Energy of a Organization

19.

Using values from Tabular array 8.2, how many Deoxyribonucleic acid molecules could be broken by the energy carried by a single electron in the beam of an erstwhile-fashioned TV tube? (These electrons were non dangerous in themselves, but they did create dangerous X-rays. Later-model tube TVs had shielding that absorbed X-rays earlier they escaped and exposed viewers.)

twenty .

If the energy in fusion bombs were used to supply the energy needs of the globe, how many of the 9-megaton diversity would be needed for a year'due south supply of energy (using data from Table 8.1)?

21.

A camera weighing 10 N falls from a small drone hovering twenty m 20m  overhead and enters free fall. What is the gravitational potential free energy change of the camera from the drone to the footing if you have a reference bespeak of (a) the ground being zilch gravitational potential energy? (b) The drone being zero gravitational potential free energy? What is the gravitational potential energy of the camera (c) before it falls from the drone and (d) after the photographic camera lands on the ground if the reference point of aught gravitational potential free energy is taken to be a second person looking out of a building 30 g 30m  from the ground?

22 .

Someone drops a fifty − grand 50−g  pebble off of a docked cruise ship, lxx.0 m 70.0m  from the water line. A person on a dock 3.0 m 3.0m  from the water line holds out a net to catch the pebble. (a) How much work is washed on the pebble by gravity during the drop? (b) What is the change in the gravitational potential energy during the driblet? If the gravitational potential energy is zero at the water line, what is the gravitational potential energy (c) when the pebble is dropped? (d) When information technology reaches the net? What if the gravitational potential free energy was 30.0 thirty.0  Joules at h2o level? (e) Find the answers to the aforementioned questions in (c) and (d).

23.

A cat'southward crinkle ball toy of mass 15 chiliad 15g  is thrown straight up with an initial speed of 3 m/due south 3m/south . Assume in this problem that air drag is negligible. (a) What is the kinetic energy of the brawl as it leaves the hand? (b) How much work is done by the gravitational force during the ball'south rise to its meridian? (c) What is the alter in the gravitational potential energy of the ball during the rising to its peak? (d) If the gravitational potential energy is taken to exist cypher at the point where it leaves your paw, what is the gravitational potential free energy when information technology reaches the maximum superlative? (e) What if the gravitational potential energy is taken to be zero at the maximum height the ball reaches, what would the gravitational potential energy exist when it leaves the hand? (f) What is the maximum height the ball reaches?

8.2 Bourgeois and Non-Conservative Forces

24 .

A force F ( ten ) = ( 3.0 / 10 ) N F(x)=(3.0/x)N  acts on a particle as information technology moves forth the positivex-axis. (a) How much work does the strength do on the particle as information technology moves from x = 2.0 m x=2.0m  to x = 5.0 m? x=five.0m?  (b) Picking a user-friendly reference point of the potential energy to be naught at x = ∞ , x=∞,  notice the potential free energy for this forcefulness.

25.

A force F ( x ) = ( −5.0 10 2 + 7.0 ten ) North F(x)=(−v.0×2+7.0x)N  acts on a particle. (a) How much work does the force do on the particle equally it moves from x = 2.0 m x=2.0m  to x = v.0 m? x=5.0m?  (b) Picking a convenient reference point of the potential energy to be zero at x = ∞ , x=∞,  find the potential energy for this force.

26 .

Find the force corresponding to the potential energy U ( x ) = − a / x + b / 10 2 . U(10)=−a/ten+b/x2.

27.

The potential free energy function for either i of the two atoms in a diatomic molecule is often approximated by U ( x ) = − a / ten 12 − b / 10 6 U(ten)=−a/x12−b/x6  wherex is the distance between the atoms. (a) At what distance of seperation does the potential energy have a local minimum (non at x = ∞ ) ? x=∞)?  (b) What is the strength on an atom at this separation? (c) How does the force vary with the separation distance?

28 .

A particle of mass ii.0 kg 2.0kg  moves under the influence of the force F ( x ) = ( 3 / 10 √ ) North . F(x)=(3/x)N.  If its speed at x = 2.0 m 10=2.0m  is v = 6.0 m/s, v=6.0m/s,  what is its speed at ten = seven.0 m? x=seven.0m?

29.

A particle of mass 2.0 kg 2.0kg  moves under the influence of the force F ( x ) = ( −5 ten two + seven x ) N . F(x)=(−5×two+7x)North.  If its speed at x = −4.0 m x=−iv.0m  is v = xx.0 m/south, v=20.0m/s,  what is its speed at x = iv.0 1000 ? x=4.0m?

30 .

A crate on rollers is beingness pushed without frictional loss of free energy across the floor of a freight car (run into the following figure). The motorcar is moving to the right with a constant speed v 0 . v0.  If the crate starts at balance relative to the freight auto, then from the work-energy theorem, F d = k 5 2 / 2 , Fd=mv2/2,  whered, the distance the crate moves, andv, the speed of the crate, are both measured relative to the freight car. (a) To an observer at rest beside the tracks, what distance d ′ d′  is the crate pushed when information technology moves the distanced in the machine? (b) What are the crate's initial and concluding speeds v 0 ′ v0′  and v ′ v′  as measured by the observer abreast the tracks? (c) Bear witness that F d ′ = grand ( v ′ ) 2 / 2 − m ( v ′ 0 ) ii / two Fd′=m(v′)2/2−g(v′0)two/two  and, consequently, that work is equal to the change in kinetic energy in both reference systems.

8.3 Conservation of Free energy

31.

A boy throws a brawl of mass 0.25 kg 0.25kg  straight upward with an initial speed of 20 m / s 20m/south  When the ball returns to the boy, its speed is 17 chiliad / s 17m/s  How much much work does air resistance do on the ball during its flight?

32 .

A mouse of mass 200 one thousand falls 100 one thousand down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its autumn, how much work is done on the mouse by air resistance?

33.

Using free energy considerations and assuming negligible air resistance, bear witness that a rock thrown from a bridge 20.0 m higher up h2o with an initial speed of 15.0 m/s strikes the water with a speed of 24.eight m/s contained of the direction thrown. (Hint:show that K i + U i = K f + U f ) Ki+Ui=Kf+Uf)

34 .

A 1.0-kg ball at the end of a 2.0-m cord swings in a vertical aeroplane. At its lowest betoken the brawl is moving with a speed of 10 m/s. (a) What is its speed at the top of its path? (b) What is the tension in the string when the ball is at the bottom and at the top of its path?

35.

Ignoring details associated with friction, extra forces exerted by arm and leg muscles, and other factors, nosotros can consider a pole vault as the conversion of an athlete's running kinetic energy to gravitational potential energy. If an athlete is to lift his torso 4.8 m during a vault, what speed must he take when he plants his pole?

36 .

Tarzan grabs a vine hanging vertically from a alpine tree when he is running at 9.0 m / s . 9.0m/s.  (a) How high can he swing upward? (b) Does the length of the vine affect this height?

37.

Assume that the strength of a bow on an arrow behaves similar the spring force. In aiming the arrow, an archer pulls the bow dorsum 50 cm and holds it in position with a force of 150 Northward 150N . If the mass of the pointer is 50 chiliad 50g  and the "spring" is massless, what is the speed of the arrow immediately after it leaves the bow?

38 .

A 100 − kg 100−kg  man is skiing beyond level ground at a speed of 8.0 one thousand/southward 8.0m/s  when he comes to the minor slope i.8 thousand higher than ground level shown in the following figure. (a) If the skier coasts up the colina, what is his speed when he reaches the top plateau? Presume friction between the snow and skis is negligible. (b) What is his speed when he reaches the upper level if an fourscore − N 80−N  frictional force acts on the skis?

39.

A sled of mass lxx kg starts from rest and slides down a 10 ° 10°  incline lxxx m 80m  long. It then travels for 20 k horizontally before starting back up an 8 ° 8°  incline. Information technology travels 80 m along this incline before coming to rest. What is the net work done on the sled by friction?

40 .

A daughter on a skateboard (total mass of 40 kg) is moving at a speed of 10 thousand/s at the bottom of a long ramp. The ramp is inclined at xx ° 20°  with respect to the horizontal. If she travels 14.2 mupward along the ramp earlier stopping, what is the cyberspace frictional strength on her?

41.

A baseball game of mass 0.25 kg is hit at dwelling plate with a speed of twoscore thou/southward. When it lands in a seat in the left-field bleachers a horizontal distance 120 m from home plate, it is moving at 30 m/s. If the brawl lands 20 grand to a higher place the spot where it was hit, how much work is done on it by air resistance?

42 .

A small block of massm slides without friction effectually the loop-the-loop apparatus shown below. (a) If the block starts from rest atA, what is its speed atB? (b) What is the forcefulness of the track on the block atB?

43.

The massless bound of a spring gun has a force constant k = 12 N/cm . thou=12N/cm.  When the gun is aimed vertically, a 15-g projectile is shot to a height of 5.0 one thousand above the terminate of the expanded spring. (See below.) How much was the spring compressed initially?

44 .

A small ball is tied to a cord and set rotating with negligible friction in a vertical circle. Bear witness that the tension in the string at the bottom of the circumvolve exceeds that at the acme of the circle by eight times the weight of the ball. Assume the brawl'due south speed is nix equally it sails over the top of the circumvolve and there is no additional energy added to the ball during rotation.

8.4 Potential Energy Diagrams and Stability

45.

A mysterious constant strength of 10 Northward acts horizontally on everything. The direction of the strength is found to exist e'er pointed toward a wall in a big hall. Find the potential energy of a particle due to this forcefulness when it is at a altitudeten from the wall, assuming the potential energy at the wall to exist zip.

46 .

A single forcefulness F ( x ) = −4.0 x F(x)=−four.0x  (in newtons) acts on a 1.0-kg body. When x = iii.5 chiliad, ten=3.5m,  the speed of the body is 4.0 m/s. What is its speed at x = two.0 m? x=2.0m?

47.

A particle of mass 4.0 kg is constrained to move along theten-centrality under a single force F ( x ) = − c x three , F(ten)=−cx3,  where c = 8.0 N/m 3 . c=8.0N/m3. The particle's speed atA, where x A = one.0 grand, xA=i.0m,  is half-dozen.0 chiliad/s. What is its speed atB, where x B = −ii.0 thou? xB=−2.0m?

48 .

The force on a particle of mass 2.0 kg varies with position according to F ( x ) = −3.0 ten two F(10)=−iii.0×two  (x in meters,F(x) in newtons). The particle's velocity at x = 2.0 m x=2.0m  is 5.0 m/southward. Calculate the mechanical free energy of the particle using (a) the origin as the reference point and (b) x = iv.0 m x=iv.0m  as the reference indicate. (c) Detect the particle's velocity at x = 1.0 thou . x=1.0m.  Do this part of the trouble for each reference betoken.

49.

A 4.0-kg particle moving along thex-axis is acted upon by the force whose functional form appears below. The velocity of the particle at x = 0 10=0  is 5 = 6.0 grand/s . five=6.0m/due south.  Notice the particle's speed at x = ( a ) 2.0 m , ( b ) 4.0 m , ( c ) ten.0 m , ( d ) x=(a)two.0m,(b)4.0m,(c)x.0m,(d)  Does the particle plow effectually at some signal and head dorsum toward the origin? (e) Repeat part (d) if v = two.0 m/south at x = 0 . v=ii.0m/satx=0.

50 .

A particle of mass 0.50 kg moves along thex-axis with a potential free energy whose dependence onten is shown below. (a) What is the force on the particle at x = 2.0 , five.0 , eight.0 , and 10=2.0,5.0,8.0,and  12 g? (b) If the total mechanical energyE of the particle is −6.0 J, what are the minimum and maximum positions of the particle? (c) What are these positions if East = 2.0 J? E=2.0J?  (d) If Eastward = 16 J E=16J , what are the speeds of the particle at the positions listed in function (a)?

51.

(a) Sketch a graph of the potential free energy role U ( ten ) = k x 2 / two + A e − α x 2 , U(10)=kx2/ii+Ae−αx2,  where grand , A , and α thousand,A,andα  are constants. (b) What is the force corresponding to this potential energy? (c) Suppose a particle of massg moving with this potential energy has a velocity v a va  when its position is x = a x=a . Show that the particle does non laissez passer through the origin unless

A ≤ m v a ii + one thousand a 2 two ( i − eastward − α a two ) . A≤mva2+ka22(one−eastward−αa2).

8.5 Sources of Energy

52 .

In the cartoon moving picturePocahontas, Pocahontas runs to the border of a cliff and jumps off, showcasing the fun side of her personality. (a) If she is running at 3.0 m/s before jumping off the cliff and she hits the h2o at the bottom of the cliff at twenty.0 m/due south, how high is the cliff? Assume negligible air drag in this cartoon. (b) If she jumped off the aforementioned cliff from a standstill, how fast would she be falling right before she hit the h2o?

53.

In the reality television receiver bear witness "Amazing Race", a contestant is firing 12-kg watermelons from a slingshot to striking targets down the field. The slingshot is pulled back 1.5 m and the watermelon is considered to be at ground level. The launch betoken is 0.iii m from the basis and the targets are 10 k horizontally away. Calculate the spring constant of the slingshot.

54 .

In theBack to the Future movies, a DeLorean car of mass 1230 kg travels at 88 miles per hr to venture back to the future. (a) What is the kinetic energy of the DeLorian? (b) What spring constant would be needed to stop this DeLorean in a distance of 0.1m?

55.

In theHunger Games film, Katniss Everdeen fires a 0.0200-kg pointer from ground level to pierce an apple upwards on a stage. The jump constant of the bow is 330 N/g and she pulls the pointer dorsum a altitude of 0.55 m. The apple on the stage is 5.00 chiliad higher than the launching point of the arrow. At what speed does the arrow (a) leave the bow? (b) strike the apple?

56 .

In a "Top Fail" video, 2 women run at each other and collide by hitting exercise balls together. If each woman has a mass of 50 kg, which includes the practice brawl, and i woman runs to the right at 2.0 g/s and the other is running toward her at i.0 thousand/s, (a) how much total kinetic energy is there in the system? (b) If free energy is conserved subsequently the collision and each exercise brawl has a mass of 2.0 kg, how fast would the balls fly off toward the camera?

57.

In a Coyote/Road Runner cartoon clip, a jump expands quickly and sends the coyote into a rock. If the spring extended 5 m and sent the coyote of mass xx kg to a speed of 15 grand/s, (a) what is the spring abiding of this leap? (b) If the coyote were sent vertically into the air with the energy given to him by the spring, how loftier could he become if there were no non-conservative forces?

58 .

In an iconic movie scene, Forrest Gump runs around the country. If he is running at a constant speed of iii m/s, would it take him more or less free energy to run uphill or downhill and why?

59.

In the movieMonty Python and the Holy Grail a cow is catapulted from the acme of a castle wall over to the people down below. The gravitational potential energy is set to nix at footing level. The cow is launched from a bound of spring constant 1.ane × 10 four N/m i.one×104N/m  that is expanded 0.v 1000 from equilibrium. If the castle is ix.one one thousand tall and the mass of the cow is 110 kg, (a) what is the gravitational potential energy of the cow at the tiptop of the castle? (b) What is the elastic spring free energy of the cow before the catapult is released? (c) What is the speed of the cow right before information technology lands on the ground?

sixty .

A 60.0-kg skier with an initial speed of 12.0 m/s coasts up a 2.l-m loftier rise as shown. Discover her final speed at the meridian, given that the coefficient of friction betwixt her skis and the snowfall is 0.lxxx.

61.

(a) How high a hill tin can a car declension upwards (engines disengaged) if work done by friction is negligible and its initial speed is 110 km/h? (b) If, in authenticity, a 750-kg automobile with an initial speed of 110 km/h is observed to coast up a loma to a height 22.0 m above its starting signal, how much thermal energy was generated past friction? (c) What is the boilerplate force of friction if the loma has a slope of ii.5 ° ii.5°  to a higher place the horizontal?

62 .

A 5.00 × 10 five -kg 5.00×105-kg  subway train is brought to a stop from a speed of 0.500 yard/s in 0.400 m by a large spring bumper at the finish of its rail. What is the spring constantk of the spring?

63.

A pogo stick has a spring with a leap constant of 2.5 × x four N/g, 2.v×104N/one thousand,  which can exist compressed 12.0 cm. To what maximum pinnacle from the uncompressed spring can a child jump on the stick using only the energy in the spring, if the child and stick have a full mass of 40 kg?

64 .

A cake of mass 500 g is fastened to a spring of spring constant 80 N/m (see the following figure). The other finish of the leap is attached to a support while the mass rests on a rough surface with a coefficient of friction of 0.20 that is inclined at angle of 30 ° . 30°.  The block is pushed along the surface till the jump compresses by 10 cm and is then released from rest. (a) How much potential energy was stored in the block-spring-support arrangement when the block was just released? (b) Make up one's mind the speed of the block when it crosses the signal when the leap is neither compressed nor stretched. (c) Determine the position of the block where it just comes to remainder on its way up the incline.

65.

A block of mass 200 thousand is fastened at the cease of a massless jump of spring constant 50 N/one thousand. The other finish of the bound is fastened to the ceiling and the mass is released at a height considered to exist where the gravitational potential free energy is cypher. (a) What is the internet potential energy of the block at the instant the block is at the lowest point? (b) What is the net potential energy of the cake at the midpoint of its descent? (c) What is the speed of the block at the midpoint of its descent?

66 .

A T-shirt cannon launches a shirt at 5.00 g/s from a platform height of 3.00 k from ground level. How fast will the shirt be traveling if information technology is caught by someone whose hands are (a) 1.00 m from ground level? (b) iv.00 m from ground level? Fail air drag.

67.

A child (32 kg) jumps upwards and down on a trampoline. The trampoline exerts a spring restoring force on the child with a constant of 5000 North/m. At the highest point of the bounce, the child is one.0 m above the level surface of the trampoline. What is the compression altitude of the trampoline? Neglect the bending of the legs or whatsoever transfer of energy of the child into the trampoline while jumping.

68 .

Shown below is a box of mass grand ane m1  that sits on a frictionless incline at an bending above the horizontal θ θ . This box is connected by a relatively massless string, over a frictionless pulley, and finally connected to a box at balance over the ledge, labeled m 2 m2 . If 1000 1 m1  and 1000 ii m2  are a meridianh above the ground and m 2 >> m 1 m2>>m1 : (a) What is the initial gravitational potential energy of the system? (b) What is the final kinetic energy of the system?

Additional Issues

69.

A massless spring with forcefulness abiding k = 200 N/m thousand=200N/m  hangs from the ceiling. A 2.0-kg block is attached to the free end of the spring and released. If the block falls 17 cm before starting back upwards, how much piece of work is done past friction during its descent?

70 .

A particle of mass ii.0 kg moves under the influence of the strength F ( 10 ) = ( −v ten 2 + 7 x ) Due north . F(x)=(−v×2+7x)N.  Suppose a frictional forcefulness also acts on the particle. If the particle'southward speed when it starts at ten = −iv.0 one thousand x=−4.0m  is 0.0 m/south and when it arrives at x = iv.0 m x=4.0m  is 9.0 m/s, how much work is done on information technology by the frictional strength between x = −4.0 m x=−four.0m  and x = 4.0 g? x=4.0m?

71.

Block ii shown below slides along a frictionless table as block 1 falls. Both blocks are attached past a frictionless pulley. Observe the speed of the blocks later they have each moved two.0 m. Assume that they start at rest and that the pulley has negligible mass. Use m 1 = ii.0 kg m1=2.0kg  and m 2 = iv.0 kg . m2=4.0kg.

72 .

A body of massm and negligible size starts from balance and slides downward the surface of a frictionless solid sphere of radiusR. (See below.) Prove that the body leaves the sphere when θ = cos −1 ( 2 / 3 ) . θ=cos−ane(ii/iii).

73.

A mysterious strength acts on all particles along a item line and always points towards a particular betokenP on the line. The magnitude of the strength on a particle increases as the cube of the distance from that indicate; that is F ∞ r 3 F∞r3 , if the distance fromP to the position of the particle isr. Letb be the proportionality abiding, and write the magnitude of the force as F = b r 3 F=br3 . Notice the potential energy of a particle subjected to this strength when the particle is at a altitudeD fromP, bold the potential free energy to be zero when the particle is atP.

74 .

An object of mass 10 kg is released at indicateA, slides to the bottom of the 30 ° 30°  incline, then collides with a horizontal massless leap, compressing information technology a maximum distance of 0.75 thousand. (Meet below.) The leap abiding is 500 G/m, the height of the incline is 2.0 m, and the horizontal surface is frictionless. (a) What is the speed of the object at the bottom of the incline? (b) What is the work of friction on the object while it is on the incline? (c) The jump recoils and sends the object dorsum toward the incline. What is the speed of the object when it reaches the base of the incline? (d) What vertical distance does it move back up the incline?

75.

Shown below is a small brawl of massm attached to a string of lengtha. A small-scale peg is located a distanceh below the point where the string is supported. If the ball is released when the string is horizontal, show thath must be greater than 3a/5 if the brawl is to swing completely around the peg.

76 .

A block leaves a frictionless inclined surfarce horizontally after dropping off past a heighth. Observe the horizontal altitudeDwhere it will land on the floor, in terms ofh,H, andgrand.

77.

A block of massm, after sliding down a frictionless incline, strikes some other block of massM that is fastened to a bound of bound constantk (see beneath). The blocks stick together upon impact and travel together. (a) Find the compression of the jump in terms ofm,M,h,g, andk when the combination comes to residue. Hint: The speed of the combined blocks m + Yard ( v 2 ) m+Chiliad(v2) is based on the speed of blockm just prior to the collision with the blockGrand (five₁) based on the equation v 2 = ( m / m ) + Grand ( v 1 ) v2=(grand/m)+M(v1) . This volition be discussed further in the chapter on Linear Momentum and Collisions. (b) The loss of kinetic energy as a event of the bonding of the 2 masses upon impact is stored in the and then-called bounden free energy of the two masses. Summate the bounden energy.

78 .

A block of mass 300 one thousand is attached to a spring of spring abiding 100 N/one thousand. The other end of the spring is attached to a back up while the block rests on a polish horizontal table and can slide freely without whatsoever friction. The block is pushed horizontally till the leap compresses by 12 cm, and then the block is released from residue. (a) How much potential energy was stored in the block-leap support system when the block was but released? (b) Make up one's mind the speed of the cake when information technology crosses the point when the spring is neither compressed nor stretched. (c) Decide the speed of the block when it has traveled a distance of 20 cm from where it was released.

79.

Consider a block of mass 0.200 kg attached to a bound of spring constant 100 North/1000. The cake is placed on a frictionless table, and the other end of the bound is attached to the wall and then that the bound is level with the table. The block is then pushed in so that the spring is compressed by x.0 cm. Find the speed of the block equally it crosses (a) the point when the spring is not stretched, (b) v.00 cm to the left of point in (a), and (c) five.00 cm to the right of betoken in (a).

80 .

A skier starts from residuum and slides downhill. What will be the speed of the skier if he drops by twenty meters in vertical height? Ignore whatsoever air resistance (which volition, in reality, be quite a lot), and any friction between the skis and the snow.

81.

Echo the preceding problem, but this fourth dimension, suppose that the work done by air resistance cannot exist ignored. Let the work done past the air resistance when the skier goes fromA toB along the given hilly path be −2000 J. The work done by air resistance is negative since the air resistance acts in the contrary direction to the displacement. Supposing the mass of the skier is 50 kg, what is the speed of the skier at pointB?

82 .

Two bodies are interacting by a conservative force. Show that the mechanical free energy of an isolated system consisting of ii bodies interacting with a conservative force is conserved. (Hint: Start by using Newton'due south third law and the definition of piece of work to find the work done on each torso by the bourgeois forcefulness.)

83.

In an amusement park, a motorcar rolls in a track every bit shown below. Discover the speed of the car atA,B, andC. Note that the piece of work done by the rolling friction is zilch since the displacement of the signal at which the rolling friction acts on the tires is momentarily at balance and therefore has a zero displacement.

84 .

A 200-k steel ball is tied to a two.00-thousand "massless" string and hung from the ceiling to brand a pendulum, and so, the brawl is brought to a position making a 30 ° 30°  angle with the vertical management and released from rest. Ignoring the effects of the air resistance, discover the speed of the ball when the cord (a) is vertically down, (b) makes an bending of twenty ° twenty°  with the vertical and (c) makes an bending of ten ° x°  with the vertical.

85.

A hockey puck is shot beyond an ice-covered swimming. Before the hockey puck was striking, the puck was at rest. After the hitting, the puck has a speed of forty m/s. The puck comes to remainder afterward going a altitude of 30 m. (a) Describe how the energy of the puck changes over time, giving the numerical values of any work or free energy involved. (b) Discover the magnitude of the internet friction force.

86 .

A projectile of mass two kg is fired with a speed of 20 m/due south at an bending of 30 ° thirty°  with respect to the horizontal. (a) Summate the initial total energy of the projectile given that the reference indicate of zero gravitational potential energy at the launch position. (b) Summate the kinetic energy at the highest vertical position of the projectile. (c) Summate the gravitational potential energy at the highest vertical position. (d) Calculate the maximum height that the projectile reaches. Compare this result by solving the aforementioned problem using your noesis of projectile motion.

87.

An arms beat is fired at a target 200 m to a higher place the footing. When the beat is 100 chiliad in the air, it has a speed of 100 thou/s. What is its speed when it hits its target? Fail air friction.

88 .

How much energy is lost to a dissipative drag force if a threescore-kg person falls at a constant speed for fifteen meters?

89.

A box slides on a frictionless surface with a total energy of fifty J. It hits a spring and compresses the jump a distance of 25 cm from equilibrium. If the same box with the same initial energy slides on a rough surface, it only compresses the spring a distance of fifteen cm, how much energy must have been lost by sliding on the crude surface?