Chapter Review Forces and Newtons Laws Part a Vocabulary Review

8 Potential Free energy and Conservation of Energy

8 Affiliate Review

Fundamental Terms

conservative strength

force that does work independent of path

conserved quantity

one that cannot be created or destroyed, simply may be transformed between dissimilar forms of itself

energy conservation

total free energy of an isolated system is constant

equilibrium point

position where the causeless conservative, internet force on a particle, given by the gradient of its potential energy curve, is nil

exact differential

is the full differential of a role and requires the apply of partial derivatives if the function involves more than one dimension

mechanical energy

sum of the kinetic and potential energies

non-conservative strength

strength that does work that depends on path

not-renewable

energy source that is non renewable, but is depleted by human consumption

potential energy

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

potential energy diagram

graph of a particle's potential energy every bit a function of position

potential energy difference

negative of the work done interim between 2 points in space

renewable

energy source that is replenished past natural processes, over human fourth dimension scales

turning bespeak

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

Central Equations

Difference of potential free energy	[latex]\Delta {U}_{AB}={U}_{B}-{U}_{A}=\text{−}{W}_{AB}[/latex]
Potential energy with respect to naught of potential energy at	[latex]{\mathbf{\overset{\to }{r}}}_{0}\Delta U=U(\mathbf{\overset{\to }{r}})-U({\mathbf{\overset{\to }{r}}}_{0})[/latex]
Gravitational potential free energy near Earth's surface	[latex]U(y)=mgy+\text{const}.[/latex]
Potential energy for an ideal spring	[latex]U(10)=\frac{ane}{2}chiliad{x}^{2}+\text{const}.[/latex]
Work done by conservative strength over a airtight path	[latex]{W}_{\text{airtight path}}=\oint {\mathbf{\overset{\to }{E}}}_{\text{cons}}\cdot d\mathbf{\overset{\to }{r}}=0[/latex]
Status for conservative strength in two dimensions	[latex](\frac{d{F}_{ten}}{dy})=(\frac{d{F}_{y}}{dx})[/latex]
Conservative force is the negative derivative of potential energy	[latex]{F}_{l}=-\frac{dU}{dl}[/latex]
Conservation of energy with no not-conservative forces	[latex]0={West}_{nc,AB}=\Delta {(K+U)}_{AB}=\Delta {E}_{AB}.[/latex]

Summary

8.one Potential Energy of a System

• For a single-particle system, the departure of potential energy is the reverse of the piece of work done by the forces interim on the particle equally it moves from one position to another.
• Since only differences of potential energy are physically meaningful, the nada of the potential free energy function tin be chosen at a user-friendly location.
• The potential energies for Earth'southward abiding gravity, about its surface, and for a Hooke's constabulary forcefulness are linear and quadratic functions of position, respectively.

8.two Bourgeois and Non-Conservative Forces

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

eight.three Conservation of Free energy

• A conserved quantity is a physical property that stays constant regardless of the path taken.
• A class of the work-energy theorem says that the modify in the mechanical energy of a particle equals the work done on it by not-conservative forces.
• If non-bourgeois forces do no work and at that place are no external forces, the mechanical energy of a particle stays constant. This is a statement of the conservation of mechanical energy and there is no alter in the total mechanical energy.
• For one-dimensional particle motility, in which the mechanical energy is constant and the potential energy is known, the particle'due south position, as a function of fourth dimension, can exist found by evaluating an integral that is derived from the conservation of mechanical free energy.

8.iv Potential Energy Diagrams and Stability

• Interpreting a i-dimensional potential energy diagram allows you to obtain qualitative, and some quantitative, information about the movement of a particle.
• At a turning point, the potential energy equals the mechanical free energy and the kinetic energy is zero, indicating that the direction of the velocity reverses there.
• The negative of the slope of the potential free energy curve, for a particle, equals the one-dimensional component of the conservative force on the particle. At an equilibrium bespeak, the gradient is zero and is a stable (unstable) equilibrium for a potential energy minimum (maximum).

eight.5 Sources of Free energy

• Free energy can exist transferred from one system to another and transformed or converted from 1 type into another. Some of the basic types of energy are kinetic, potential, thermal, and electromagnetic.
• Renewable free energy sources are those that are replenished by ongoing natural processes, over human being time scales. Examples are current of air, water, geothermal, and solar power.
• Non-renewable free energy sources are those that are depleted past consumption, over human time scales. Examples are fossil fuel and nuclear ability.

Conceptual Questions

8.1 Potential Energy of a System

one.

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

2 .

The strength exerted by a diving board is conservative, provided the internal friction is negligible. Bold friction is negligible, describe changes in the potential free energy of a diving board as a swimmer drives from it, starting but before the swimmer steps on the lath until but after his anxiety leave information technology.

3.

Describe the gravitational potential energy transfers and transformations for a javelin, starting from the point at which an athlete picks upward the javelin and catastrophe when the javelin is stuck into the footing afterwards existence thrown.

4 .

A couple of soccer balls of equal mass are kicked off the ground at the aforementioned speed merely at unlike angles. Soccer ball A is kicked off at an bending slightly above the horizontal, whereas ball B is kicked slightly below the vertical. How do each of the post-obit compare for brawl A and brawl B? (a) The initial kinetic energy and (b) the change in gravitational potential free energy from the footing to the highest point? If the energy in part (a) differs from part (b), explicate why there is a difference between the 2 energies.

v.

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

6 .

Ii people notice a leaf falling from a tree. One person is standing on a ladder and the other is on the footing. If each person were to compare the energy of the leaf observed, would each person detect the post-obit to exist the same or unlike for the leaf, from the indicate where information technology falls off the tree to when it hits the ground: (a) the kinetic free energy of the leaf; (b) the change in gravitational potential energy; (c) the final gravitational potential energy?

eight.2 Conservative and Non-Conservative Forces

7.

What is the concrete meaning of a non-conservative force?

eight .

A bottle rocket is shot straight upwards in the air with a speed 30 m/s 30m/southward . If the air resistance is ignored, the bottle would go upwards to a height of approximately 46 thou 46m . However, the rocket goes up to only 35 m 35m  earlier returning to the basis. What happened? Explain, giving only a qualitative response.

ix.

An external forcefulness acts on a particle during a trip from one signal to another and back to that same point. This particle is just effected by conservative forces. Does this particle's kinetic energy and potential free energy alter as a result of this trip?

8.iii Conservation of Energy

10 .

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

11.

Consider the post-obit scenario. A car for which friction isnot negligible accelerates from rest down a loma, running out of gasoline after a short distance (run across beneath). The driver lets the car coast further down the hill, then up and over a small-scale crest. He so coasts down that hill into a gas station, where he brakes to a stop and fills the tank with gasoline. Identify the forms of free energy the car has, and how they are changed and transferred in this series of events.

12 .

A dropped ball bounces to one-half its original height. Discuss the energy transformations that take place.

thirteen.

" E = K + U E=Yard+U  constant is a special example of the work-free energy theorem." Discuss this argument.

14 .

In a mutual physics demonstration, a bowling ball is suspended from the ceiling by a rope.

The professor pulls the ball abroad from its equilibrium position and holds information technology side by side to his nose, as shown below. He releases the ball and so that it swings directly abroad from him. Does he get struck by the ball on its render swing? What is he trying to prove in this demonstration?

15.

A child jumps up and down on a bed, reaching a higher height afterward each bounce. Explain how the child can increase his maximum gravitational potential energy with each bounciness.

16 .

Tin can a non-conservative force increase the mechanical energy of the arrangement?

17.

Neglecting air resistance, how much would I have to raise the vertical height if I wanted to double the impact speed of a falling object?

xviii .

A box is dropped onto a spring at its equilibrium position. The leap compresses with the box attached and comes to rest. Since the spring is in the vertical position, does the modify in the gravitational potential free energy of the box while the jump is compressing demand to exist considered in this problem?

Problems

8.one Potential Energy of a System

19.

Using values from Table viii.2, how many DNA molecules could be cleaved by the energy carried by a single electron in the beam of an quondam-fashioned Tv tube? (These electrons were not 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.)

xx .

If the energy in fusion bombs were used to supply the free energy needs of the world, how many of the ix-megaton variety would exist needed for a year'southward supply of energy (using data from Table 8.1)?

21.

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

22 .

Someone drops a 50 − grand 50−chiliad  pebble off of a docked cruise transport, lxx.0 m 70.0m  from the h2o line. A person on a dock 3.0 m iii.0m  from the water line holds out a net to catch the pebble. (a) How much work is done on the pebble by gravity during the drop? (b) What is the change in the gravitational potential energy during the drib? If the gravitational potential energy is nix at the h2o line, what is the gravitational potential free energy (c) when the pebble is dropped? (d) When it reaches the cyberspace? What if the gravitational potential free energy was 30.0 30.0  Joules at water level? (eastward) Find the answers to the same questions in (c) and (d).

23.

A cat's crinkle ball toy of mass xv m 15g  is thrown straight upward with an initial speed of 3 m/s 3m/due south . Assume in this problem that air drag is negligible. (a) What is the kinetic energy of the ball equally information technology leaves the hand? (b) How much piece of work is done past the gravitational force during the ball'due south ascension to its tiptop? (c) What is the modify in the gravitational potential energy of the ball during the ascension to its peak? (d) If the gravitational potential energy is taken to exist zero at the bespeak where it leaves your hand, what is the gravitational potential energy when it reaches the maximum height? (east) 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 information technology leaves the hand? (f) What is the maximum tiptop the ball reaches?

eight.2 Bourgeois and Not-Conservative Forces

24 .

A force F ( x ) = ( 3.0 / x ) Northward F(ten)=(3.0/10)N  acts on a particle every bit it moves along the positiveten-axis. (a) How much work does the force do on the particle as it moves from x = 2.0 m x=ii.0m  to x = 5.0 m? x=5.0m?  (b) Picking a convenient reference indicate of the potential free energy to exist zero at 10 = ∞ , ten=∞,  find the potential energy for this force.

25.

A force F ( ten ) = ( −5.0 x ii + vii.0 10 ) North F(ten)=(−5.0×2+7.0x)N  acts on a particle. (a) How much work does the force do on the particle as it moves from 10 = 2.0 m x=ii.0m  to x = v.0 one thousand? ten=five.0m?  (b) Picking a convenient reference betoken of the potential free energy to be naught at x = ∞ , x=∞,  notice the potential energy for this force.

26 .

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

27.

The potential energy role for either one of the two atoms in a diatomic molecule is often approximated by U ( x ) = − a / x 12 − b / x 6 U(x)=−a/x12−b/x6  whereten is the altitude between the atoms. (a) At what distance of seperation does the potential energy have a local minimum (not at x = ∞ ) ? x=∞)?  (b) What is the force on an atom at this separation? (c) How does the forcefulness vary with the separation altitude?

28 .

A particle of mass 2.0 kg 2.0kg  moves nether the influence of the forcefulness F ( x ) = ( 3 / x √ ) N . F(x)=(three/10)Northward.  If its speed at x = two.0 m x=2.0m  is v = vi.0 k/due south, v=vi.0m/s,  what is its speed at 10 = 7.0 yard? x=7.0m?

29.

A particle of mass 2.0 kg 2.0kg  moves under the influence of the force F ( 10 ) = ( −5 x 2 + 7 x ) Northward . F(x)=(−5×2+7x)North.  If its speed at 10 = −4.0 m x=−four.0m  is v = 20.0 m/s, 5=20.0m/s,  what is its speed at 10 = four.0 m ? ten=4.0m?

30 .

A crate on rollers is being pushed without frictional loss of energy across the floor of a freight car (run into the following figure). The car is moving to the right with a constant speed v 0 . v0.  If the crate starts at rest relative to the freight car, so from the work-energy theorem, F d = m v two / 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 it moves the distanced in the motorcar? (b) What are the crate's initial and last speeds v 0 ′ v0′  and v ′ v′  every bit measured by the observer beside the tracks? (c) Prove that F d ′ = m ( v ′ ) 2 / ii − g ( five ′ 0 ) 2 / two Fd′=m(v′)2/2−m(v′0)two/2  and, consequently, that work is equal to the change in kinetic energy in both reference systems.

8.iii Conservation of Free energy

31.

A male child throws a ball of mass 0.25 kg 0.25kg  straight upwards with an initial speed of xx g / s 20m/s  When the ball returns to the boy, its speed is 17 thou / s 17m/due south  How much much work does air resistance practice on the ball during its flying?

32 .

A mouse of mass 200 g falls 100 chiliad downward a vertical mine shaft and lands at the bottom with a speed of 8.0 k/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, evidence that a stone thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/southward strikes the water with a speed of 24.8 m/s independent 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 ii.0-m cord swings in a vertical aeroplane. At its everyman betoken the ball is moving with a speed of 10 m/southward. (a) What is its speed at the top of its path? (b) What is the tension in the string when the brawl 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, we 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 body 4.8 m during a vault, what speed must he have when he plants his pole?

36 .

Tarzan grabs a vine hanging vertically from a tall tree when he is running at 9.0 grand / 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 force of a bow on an arrow behaves similar the bound force. In aiming the pointer, an archer pulls the bow back l cm and holds it in position with a force of 150 Northward 150N . If the mass of the pointer is l m 50g  and the "spring" is massless, what is the speed of the arrow immediately subsequently it leaves the bow?

38 .

A 100 − kg 100−kg  man is skiing across level ground at a speed of 8.0 yard/s 8.0m/south  when he comes to the small slope ane.8 m higher than basis level shown in the following effigy. (a) If the skier coasts up the hill, what is his speed when he reaches the top plateau? Assume friction between the snow and skis is negligible. (b) What is his speed when he reaches the upper level if an 80 − Due north eighty−N  frictional force acts on the skis?

39.

A sled of mass 70 kg starts from residue and slides downward a 10 ° 10°  incline 80 m 80m  long. Information technology then travels for 20 1000 horizontally before starting back up an 8 ° eight°  incline. Information technology travels 80 yard along this incline before coming to residuum. What is the cyberspace work done on the sled past friction?

40 .

A girl on a skateboard (total mass of 40 kg) is moving at a speed of x m/s at the lesser of a long ramp. The ramp is inclined at 20 ° 20°  with respect to the horizontal. If she travels 14.2 mupward forth the ramp before stopping, what is the net frictional force on her?

41.

A baseball of mass 0.25 kg is hitting at home plate with a speed of 40 m/s. When information technology lands in a seat in the left-field bleachers a horizontal distance 120 thou from home plate, it is moving at 30 chiliad/due south. If the ball lands 20 m to a higher place the spot where it was hitting, how much work is done on it by air resistance?

42 .

A small block of mass1000 slides without friction around the loop-the-loop appliance shown below. (a) If the block starts from rest atA, what is its speed atB? (b) What is the force of the runway on the block atB?

43.

The massless leap of a spring gun has a force constant k = 12 N/cm . m=12N/cm.  When the gun is aimed vertically, a 15-k projectile is shot to a superlative of 5.0 k to a higher place the stop of the expanded jump. (Come across below.) How much was the jump compressed initially?

44 .

A pocket-sized ball is tied to a string and gear up rotating with negligible friction in a vertical circle. Prove that the tension in the string at the bottom of the circle exceeds that at the pinnacle of the circle past eight times the weight of the brawl. Presume the ball's speed is zero as information technology sails over the height of the circle and in that location is no additional energy added to the ball during rotation.

8.four Potential Energy Diagrams and Stability

45.

A mysterious abiding strength of 10 N acts horizontally on everything. The direction of the force is found to be e'er pointed toward a wall in a big hall. Find the potential energy of a particle due to this force when it is at a distancex from the wall, bold the potential energy at the wall to be zero.

46 .

A single force F ( x ) = −4.0 ten F(10)=−4.0x  (in newtons) acts on a 1.0-kg body. When x = iii.5 thousand, x=3.5m,  the speed of the trunk is 4.0 thou/south. What is its speed at x = two.0 one thousand? x=2.0m?

47.

A particle of mass four.0 kg is constrained to move forth thex-centrality under a single force F ( x ) = − c x 3 , F(ten)=−cx3,  where c = viii.0 N/m 3 . c=8.0N/m3. The particle's speed atA, where ten A = ane.0 thou, xA=i.0m,  is six.0 thousand/s. What is its speed atB, where x B = −2.0 m? xB=−2.0m?

48 .

The force on a particle of mass 2.0 kg varies with position according to F ( 10 ) = −3.0 x 2 F(x)=−3.0×two  (x in meters,F(x) in newtons). The particle's velocity at 10 = 2.0 one thousand x=2.0m  is five.0 m/s. Calculate the mechanical energy of the particle using (a) the origin as the reference point and (b) 10 = 4.0 m x=4.0m  as the reference point. (c) Find the particle's velocity at x = 1.0 grand . x=ane.0m.  Do this part of the problem for each reference signal.

49.

A four.0-kg particle moving along thex-axis is acted upon by the forcefulness whose functional form appears beneath. The velocity of the particle at ten = 0 x=0  is v = half-dozen.0 g/s . v=6.0m/s.  Find the particle'south speed at x = ( a ) ii.0 m , ( b ) 4.0 grand , ( c ) x.0 yard , ( d ) 10=(a)2.0m,(b)4.0m,(c)x.0m,(d)  Does the particle turn around at some signal and caput back toward the origin? (eastward) Echo office (d) if v = 2.0 g/southward at x = 0 . v=2.0m/satx=0.

50 .

A particle of mass 0.50 kg moves forth theten-centrality with a potential free energy whose dependence on10 is shown below. (a) What is the force on the particle at ten = 2.0 , five.0 , viii.0 , and x=ii.0,5.0,eight.0,and  12 g? (b) If the total mechanical energyE of the particle is −six.0 J, what are the minimum and maximum positions of the particle? (c) What are these positions if E = ii.0 J? East=2.0J?  (d) If Due east = sixteen J East=16J , what are the speeds of the particle at the positions listed in part (a)?

51.

(a) Sketch a graph of the potential energy function U ( 10 ) = thou x 2 / 2 + A due east − α x 2 , U(x)=kx2/2+Ae−αx2,  where k , A , and α one thousand,A,andα  are constants. (b) What is the force respective to this potential energy? (c) Suppose a particle of massm moving with this potential energy has a velocity v a va  when its position is x = a x=a . Show that the particle does not pass through the origin unless

A ≤ 1000 five a 2 + k a ii 2 ( i − east − α a 2 ) . A≤mva2+ka22(1−e−αa2).

viii.5 Sources of Energy

52 .

In the cartoon moviePocahontas, Pocahontas runs to the edge of a cliff and jumps off, showcasing the fun side of her personality. (a) If she is running at 3.0 m/s earlier jumping off the cliff and she hits the h2o at the bottom of the cliff at 20.0 m/southward, 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 striking the water?

53.

In the reality television testify "Amazing Race", a contestant is firing 12-kg watermelons from a slingshot to striking targets down the field. The slingshot is pulled back i.5 m and the watermelon is considered to be at footing level. The launch point is 0.iii 1000 from the ground and the targets are x m 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 time to come. (a) What is the kinetic free energy of the DeLorian? (b) What jump abiding would be needed to stop this DeLorean in a altitude of 0.1m?

55.

In theHunger Games moving-picture show, Katniss Everdeen fires a 0.0200-kg arrow from ground level to pierce an apple upward on a stage. The spring constant of the bow is 330 North/thousand and she pulls the pointer back a altitude of 0.55 m. The apple tree on the stage is 5.00 m higher than the launching indicate of the arrow. At what speed does the arrow (a) get out the bow? (b) strike the apple?

56 .

In a "Top Fail" video, two women run at each other and collide by hitting exercise balls together. If each woman has a mass of 50 kg, which includes the exercise brawl, and one woman runs to the right at ii.0 thou/s and the other is running toward her at 1.0 thou/south, (a) how much total kinetic energy is there in the system? (b) If energy is conserved after the collision and each practice ball has a mass of 2.0 kg, how fast would the assurance fly off toward the photographic camera?

57.

In a Coyote/Road Runner cartoon prune, a spring expands quickly and sends the coyote into a rock. If the jump extended 5 thou and sent the coyote of mass xx kg to a speed of 15 m/s, (a) what is the spring abiding of this spring? (b) If the coyote were sent vertically into the air with the energy given to him past the jump, how loftier could he become if in that location were no not-conservative forces?

58 .

In an iconic movie scene, Forrest Gump runs effectually the country. If he is running at a constant speed of 3 one thousand/s, would it have 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 peak of a castle wall over to the people down below. The gravitational potential free energy is prepare to nil at footing level. The cow is launched from a spring of leap constant 1.one × ten iv N/m 1.1×104N/grand  that is expanded 0.five m from equilibrium. If the castle is ix.1 m alpine and the mass of the moo-cow is 110 kg, (a) what is the gravitational potential energy of the moo-cow at the acme of the castle? (b) What is the elastic spring energy of the moo-cow before the catapult is released? (c) What is the speed of the cow right earlier it lands on the ground?

60 .

A 60.0-kg skier with an initial speed of 12.0 grand/s coasts upwardly a 2.50-grand high rise as shown. Find her final speed at the top, given that the coefficient of friction between her skis and the snow is 0.eighty.

61.

(a) How loftier a hill tin a car coast up (engines disengaged) if piece of work done by friction is negligible and its initial speed is 110 km/h? (b) If, in actuality, a 750-kg car with an initial speed of 110 km/h is observed to coast upward a hill to a height 22.0 m above its starting signal, how much thermal energy was generated by friction? (c) What is the average force of friction if the hill has a slope of ii.5 ° two.5°  above the horizontal?

62 .

A 5.00 × 10 5 -kg v.00×105-kg  subway railroad train is brought to a stop from a speed of 0.500 thousand/southward in 0.400 m by a large spring bumper at the cease of its rail. What is the spring constantk of the spring?

63.

A pogo stick has a jump with a spring constant of 2.5 × 10 iv N/one thousand, 2.5×104N/m,  which can be compressed 12.0 cm. To what maximum pinnacle from the uncompressed spring can a child leap on the stick using only the free energy in the spring, if the child and stick have a full mass of 40 kg?

64 .

A block of mass 500 g is attached to a spring of spring constant 80 N/m (see the post-obit figure). The other end of the leap is fastened to a back up while the mass rests on a rough surface with a coefficient of friction of 0.xx that is inclined at angle of 30 ° . 30°.  The block is pushed along the surface till the spring compresses by ten cm and is and so released from rest. (a) How much potential energy was stored in the block-spring-back up system when the block was merely released? (b) Make up one's mind the speed of the block when it crosses the point when the spring is neither compressed nor stretched. (c) Decide the position of the block where it just comes to rest on its style upwardly the incline.

65.

A block of mass 200 g is attached at the finish of a massless spring of spring constant 50 N/chiliad. The other terminate of the spring is attached to the ceiling and the mass is released at a height considered to be where the gravitational potential free energy is zero. (a) What is the net potential energy of the cake at the instant the block is at the lowest signal? (b) What is the internet 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 five.00 grand/s from a platform height of three.00 m 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 footing level? (b) 4.00 grand from footing level? Neglect air drag.

67.

A child (32 kg) jumps upwards and down on a trampoline. The trampoline exerts a leap restoring forcefulness on the child with a abiding of 5000 N/k. At the highest betoken of the bounciness, the kid is 1.0 g above the level surface of the trampoline. What is the compression distance of the trampoline? Fail 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 one thousand one m1  that sits on a frictionless incline at an bending above the horizontal θ θ . This box is connected past a relatively massless string, over a frictionless caster, and finally connected to a box at rest over the ledge, labeled m ii m2 . If g 1 m1  and thousand ii m2  are a heighth to a higher place the ground and chiliad 2 >> m 1 m2>>m1 : (a) What is the initial gravitational potential energy of the organization? (b) What is the final kinetic energy of the system?

Additional Problems

69.

A massless spring with forcefulness constant k = 200 N/m 1000=200N/m  hangs from the ceiling. A 2.0-kg block is fastened to the gratuitous cease of the spring and released. If the block falls 17 cm before starting back upward, how much work is done by friction during its descent?

70 .

A particle of mass 2.0 kg moves under the influence of the force F ( x ) = ( −v x 2 + 7 x ) North . F(x)=(−5×2+7x)North.  Suppose a frictional forcefulness as well acts on the particle. If the particle's speed when information technology starts at 10 = −four.0 m x=−four.0m  is 0.0 grand/s and when it arrives at x = 4.0 m ten=4.0m  is 9.0 m/southward, how much piece of work is done on it by the frictional force between 10 = −4.0 m x=−4.0m  and x = 4.0 thou? x=4.0m?

71.

Block 2 shown below slides along a frictionless table as block one falls. Both blocks are attached by a frictionless caster. Find the speed of the blocks later they have each moved 2.0 thou. Presume that they start at rest and that the pulley has negligible mass. Apply 1000 1 = ii.0 kg m1=two.0kg  and thousand two = 4.0 kg . m2=4.0kg.

72 .

A torso of massg and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radiusR. (Come across beneath.) Prove that the torso leaves the sphere when θ = cos −1 ( 2 / 3 ) . θ=cos−1(2/3).

73.

A mysterious force acts on all particles along a particular line and always points towards a particular pointP on the line. The magnitude of the forcefulness on a particle increases equally the cube of the altitude from that point; that is F ∞ r 3 F∞r3 , if the altitude fromP to the position of the particle isr. Allowb be the proportionality constant, and write the magnitude of the strength as F = b r 3 F=br3 . Find the potential energy of a particle subjected to this strength when the particle is at a altitudeD fromP, assuming the potential energy to be aught when the particle is atP.

74 .

An object of mass 10 kg is released at pointA, slides to the bottom of the 30 ° 30°  incline, then collides with a horizontal massless spring, compressing it a maximum distance of 0.75 g. (See below.) The spring abiding is 500 M/yard, the height of the incline is 2.0 k, 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 spring 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 information technology move back up the incline?

75.

Shown below is a small ball of massgrand fastened to a string of lengtha. A small peg is located a distanceh beneath the indicate where the string is supported. If the brawl is released when the cord is horizontal, prove thath must exist greater than 3a/5 if the ball is to swing completely around the peg.

76 .

A block leaves a frictionless inclined surfarce horizontally after dropping off by a heighth. Find the horizontal distanceDwhere information technology volition land on the floor, in terms ofh,H, andg.

77.

A block of massm, after sliding down a frictionless incline, strikes another block of massG that is fastened to a bound of spring constantchiliad (run into beneath). The blocks stick together upon impact and travel together. (a) Find the compression of the spring in terms ofk,Grand,h,g, andg when the combination comes to remainder. Hint: The speed of the combined blocks m + M ( 5 ii ) m+M(v2) is based on the speed of cakeyard just prior to the collision with the cakeM (v_ane) based on the equation v ii = ( m / m ) + M ( v 1 ) v2=(yard/grand)+M(v1) . This volition exist discussed further in the chapter on Linear Momentum and Collisions. (b) The loss of kinetic energy equally a upshot of the bonding of the 2 masses upon affect is stored in the so-called binding energy of the 2 masses. Calculate the binding energy.

78 .

A cake of mass 300 g is attached to a bound of spring abiding 100 Due north/m. The other terminate of the jump is fastened to a back up while the block rests on a polish horizontal table and tin slide freely without whatever friction. The block is pushed horizontally till the spring compresses by 12 cm, and and so the block is released from rest. (a) How much potential energy was stored in the block-spring support arrangement when the cake was just released? (b) Make up one's mind the speed of the block when information technology crosses the point when the spring is neither compressed nor stretched. (c) Decide the speed of the block when information technology has traveled a distance of 20 cm from where information technology was released.

79.

Consider a block of mass 0.200 kg attached to a leap of bound constant 100 Northward/m. The block is placed on a frictionless table, and the other end of the leap is attached to the wall so that the bound is level with the table. The block is and so pushed in so that the spring is compressed past 10.0 cm. Detect the speed of the block equally it crosses (a) the betoken when the spring is not stretched, (b) 5.00 cm to the left of bespeak in (a), and (c) 5.00 cm to the right of point in (a).

lxxx .

A skier starts from remainder and slides downhill. What will exist the speed of the skier if he drops past 20 meters in vertical height? Ignore any air resistance (which will, in reality, exist quite a lot), and any friction between the skis and the snow.

81.

Repeat the preceding problem, just this fourth dimension, suppose that the piece of work done by air resistance cannot be ignored. Let the work done by the air resistance when the skier goes fromA toB along the given hilly path exist −2000 J. The work done past air resistance is negative since the air resistance acts in the reverse 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 past a bourgeois forcefulness. Show that the mechanical energy of an isolated organization consisting of two bodies interacting with a conservative force is conserved. (Hint: Kickoff by using Newton'south 3rd law and the definition of work to find the work done on each trunk by the conservative force.)

83.

In an amusement park, a auto rolls in a track as shown below. Observe the speed of the motorcar atA,B, andC. Note that the work done past the rolling friction is zero since the displacement of the point at which the rolling friction acts on the tires is momentarily at rest and therefore has a goose egg displacement.

84 .

A 200-g steel brawl is tied to a 2.00-g "massless" cord and hung from the ceiling to make a pendulum, and so, the ball is brought to a position making a 30 ° 30°  angle with the vertical direction and released from rest. Ignoring the effects of the air resistance, find the speed of the ball when the string (a) is vertically down, (b) makes an angle of 20 ° 20°  with the vertical and (c) makes an angle of x ° x°  with the vertical.

85.

A hockey puck is shot across an ice-covered pond. Before the hockey puck was hit, the puck was at rest. Later on the hit, the puck has a speed of 40 m/s. The puck comes to residual later on going a distance of thirty m. (a) Describe how the free energy of the puck changes over fourth dimension, giving the numerical values of whatsoever piece of work or energy involved. (b) Find the magnitude of the net friction force.

86 .

A projectile of mass 2 kg is fired with a speed of twenty m/s at an angle of thirty ° 30°  with respect to the horizontal. (a) Calculate the initial total energy of the projectile given that the reference betoken of zero gravitational potential energy at the launch position. (b) Calculate 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 same problem using your cognition of projectile motion.

87.

An artillery shell is fired at a target 200 thousand above the ground. When the shell is 100 thousand in the air, it has a speed of 100 m/southward. What is its speed when it hits its target? Neglect air friction.

88 .

How much free energy is lost to a dissipative drag force if a 60-kg person falls at a constant speed for 15 meters?

89.

A box slides on a frictionless surface with a total free energy of l J. Information technology hits a bound and compresses the spring a distance of 25 cm from equilibrium. If the same box with the same initial energy slides on a rough surface, it merely compresses the spring a distance of 15 cm, how much energy must have been lost by sliding on the crude surface?

vincentungs1936.blogspot.com

Source: https://pressbooks.online.ucf.edu/phy2048tjb/chapter/8-chapter-review/

Share this post