Potential energy and conservation Potential energy and conservation of energyof energy

Chapter 8Chapter 8

I.I. Potential energy Potential energy Energy of configuration Energy of configuration

II.II. Work and potential energyWork and potential energy

III.III. Conservative / Non-conservative forcesConservative / Non-conservative forces

IV.IV. Determining potential energy values:Determining potential energy values:

- Gravitational potential energy- Gravitational potential energy - Elastic potential energy- Elastic potential energy

V. V. Conservation of mechanical energyConservation of mechanical energy

VI.VI. External work and thermal energyExternal work and thermal energy

VII.VII. External forces and internal energy changesExternal forces and internal energy changes

I.I. Potential energyPotential energy

Energy associated with the Energy associated with the arrangementarrangement of a system of of a system of objects that exert forces on one another. objects that exert forces on one another.

Units: Units: 1J 1J

Examples:Examples:

- Gravitational potential energy:Gravitational potential energy: associated with the state of associated with the state of separation between objects which can attract one another separation between objects which can attract one another via the gravitational force.via the gravitational force.

- Elastic potential energy:Elastic potential energy: associated with the state of associated with the state of compression/extension of an elastic object. compression/extension of an elastic object.

II. Work and potential energyII. Work and potential energy

If tomato If tomato risesrises gravitational gravitational force transfers energy “from” force transfers energy “from” tomato’s kinetic energy “to” the tomato’s kinetic energy “to” the gravitational potential energy of the gravitational potential energy of the tomato-Earth system.tomato-Earth system.

If tomato If tomato falls downfalls down gravitational force transfers gravitational force transfers energy energy “from”“from” the gravitational potential energy the gravitational potential energy “to”“to” the the tomato’s kinetic energy.tomato’s kinetic energy.

Potential EnergyPotential Energy

―The energy storage mechanism is called The energy storage mechanism is called potential energypotential energy

―A potential energy can only be associated A potential energy can only be associated with specific types of forceswith specific types of forces

―Potential energy is always associated with a Potential energy is always associated with a system of two or more interacting objectssystem of two or more interacting objects

Gravitational Potential EnergyGravitational Potential Energy

− Gravitational Potential Energy is associated Gravitational Potential Energy is associated with an object at a given distance above with an object at a given distance above Earth’s surfaceEarth’s surface

− Assume the object is in equilibrium or moving Assume the object is in equilibrium or moving at constant velocityat constant velocity

− The work done on the object is done by The work done on the object is done by FFappapp

and the upward displacement is and the upward displacement is

ˆy r j

Gravitational Potential EnergyGravitational Potential Energy

− The quantity The quantity mgymgy is identified as the is identified as the gravitational potential energy, gravitational potential energy, UUgg

UUgg = = mgymgy − Units are joules (Units are joules (JJ))

app

ˆ ˆ( ) b a

b a

W

W mg y y

W mgy mgy

F r

j j

Gravitational Potential EnergyGravitational Potential Energy

− The gravitational potential energy depends The gravitational potential energy depends only on the vertical height of the object above only on the vertical height of the object above Earth’s surfaceEarth’s surface

− In solving problems, you must choose a In solving problems, you must choose a reference configuration for which the reference configuration for which the gravitational potential energy is set equal to gravitational potential energy is set equal to some reference value, normally zerosome reference value, normally zero• The choice is arbitrary because you normally need The choice is arbitrary because you normally need

the the differencedifference in potential energy, which is in potential energy, which is independent of the choice of reference independent of the choice of reference configurationconfiguration

Conservation of Mechanical EnergyConservation of Mechanical Energy

• The mechanical energy of a system is the The mechanical energy of a system is the algebraic sum of the kinetic and potential algebraic sum of the kinetic and potential energies in the system energies in the system

EEmechmech = = KK + + UUgg

• The statement of Conservation of Mechanical The statement of Conservation of Mechanical Energy for an isolated system is Energy for an isolated system is

KKff + + UUff = = KKii+ + UUii

– An isolated system is one for which there are no An isolated system is one for which there are no energy transfers across the boundaryenergy transfers across the boundary

• Look at the work done by the Look at the work done by the book as it falls from some book as it falls from some height height yybb to a lower height to a lower height yyaa

WWon bookon book = = ΔΔKKbookbook

• Also, Also,

WW = = mgymgybb – – mgymgyaa

• So,So,

ΔΔKK = - = -ΔΔUUgg

Conservation of Mechanical EnergyConservation of Mechanical Energy

Elastic Potential EnergyElastic Potential Energy• Elastic Potential EnergyElastic Potential Energy is associated with a spring is associated with a spring• The force the spring exerts (on a block, for example) The force the spring exerts (on a block, for example) cancan

be mathematically modeled asbe mathematically modeled as

FFss = - = - kxkx

where where x x is the position of the block relative to its equilibrium (is the position of the block relative to its equilibrium (x=0x=0) ) position and position and kk is a positive constant called the force constant or the is a positive constant called the force constant or the spring constant.spring constant.

• The force required to stretch or compress the spring is The force required to stretch or compress the spring is proportional to the amount of stretch or compression. proportional to the amount of stretch or compression. This force law for springs is known as This force law for springs is known as Hooke’s lawHooke’s law. The . The value of value of kk is a measure of the stiffness of the spring is a measure of the stiffness of the spring

The vector form of the Hook’s law:The vector form of the Hook’s law:

where we have chosen the where we have chosen the x x axis to lie along the spring extension. axis to lie along the spring extension.

ikxiFF SSˆˆ

Elastic Potential EnergyElastic Potential Energy• The work done by an external applied force on a spring-block The work done by an external applied force on a spring-block

system issystem is

If the block undergoes an arbitrary displacement from If the block undergoes an arbitrary displacement from x=xx=xi i

to to x=xx=xf f the work done by the spring force on the block isthe work done by the spring force on the block is

– The work is equal to the difference between the initial and The work is equal to the difference between the initial and final values of elastic potential energy of the block-spring final values of elastic potential energy of the block-spring systemsystem

f

i

x

x x

SS kxdxkxidxikxrdFW0

2

max2

1)()ˆ()ˆ(

f

i

x

x

fiS kxkxdxkxW 22

2

1

2

1)(

Elastic Potential EnergyElastic Potential Energy

UUss = ½ = ½ kxkx22

• The elastic potential The elastic potential energy can be thought energy can be thought of as the energy stored of as the energy stored in the deformed springin the deformed spring

• The stored potential The stored potential energy can be energy can be converted into kinetic converted into kinetic energyenergy

Elastic Potential EnergyElastic Potential Energy• The elastic potential energy stored in a spring is The elastic potential energy stored in a spring is

zero whenever the spring is not deformed (zero whenever the spring is not deformed (UU = = 00 when when xx = 0 = 0))– The energy is stored in the spring only when the The energy is stored in the spring only when the

spring is stretched or compressedspring is stretched or compressed

• The elastic potential energy is a maximum when The elastic potential energy is a maximum when the spring has reached its maximum extension the spring has reached its maximum extension or compressionor compression

• The elastic potential energy is always positive, The elastic potential energy is always positive, xx22 will always be positive will always be positive

General:General:

- System of two or more objects.System of two or more objects.

- A force acts between a particle in the system and the rest of A force acts between a particle in the system and the rest of the system.the system.

- - When system configuration changes When system configuration changes force does work on force does work on

the object (the object (WW11) transferring energy between ) transferring energy between KEKE of the object of the object

and some other form of energy of the system.and some other form of energy of the system.

- When the configuration change is reversed When the configuration change is reversed force reverses force reverses

the energy transfer, doing the energy transfer, doing WW22..

Problem Solving Strategy – Conservation Problem Solving Strategy – Conservation of Mechanical Energyof Mechanical Energy

• Define the isolated system and the initial and Define the isolated system and the initial and final configuration of the systemfinal configuration of the system– The system may include two or more interacting The system may include two or more interacting

particlesparticles– The system may also include springs or other The system may also include springs or other

structures in which elastic potential energy can be structures in which elastic potential energy can be storedstored

– Also include all components of the system that Also include all components of the system that exert forces on each otherexert forces on each other

Problem-Solving StrategyProblem-Solving Strategy

• Identify the configuration for zero Identify the configuration for zero potential energy potential energy – Include both gravitational and elastic Include both gravitational and elastic

potential energiespotential energies– If more than one force is acting within the If more than one force is acting within the

system, write an expression for the system, write an expression for the potential energy associated with each forcepotential energy associated with each force

• If friction or air resistance is present, If friction or air resistance is present, mechanical energy of the system is not mechanical energy of the system is not conservedconserved

• Use energy with non-conservative Use energy with non-conservative forces insteadforces instead

Problem-Solving StrategyProblem-Solving Strategy

• If the mechanical energy of the system If the mechanical energy of the system is conserved, write the total energy asis conserved, write the total energy asEEii = = KKii + + UUi i for the initial configurationfor the initial configuration

EEff = = KKff + + UUff for the final configurationfor the final configuration

• Since mechanical energy is conserved, Since mechanical energy is conserved, EEii = = EEff

and you can solve for the unknown quantityand you can solve for the unknown quantity

Problem-Solving StrategyProblem-Solving Strategy

Conservation of Energy Example 1 (Drop a Ball)

• Initial conditions:Initial conditions:

EEii = = KKii + + UUii = = mghmgh– The ball is dropped, so The ball is dropped, so KKii

= 0= 0• The configuration for The configuration for

zero potential energy is zero potential energy is the groundthe ground

• Conservation rules Conservation rules applied at some point applied at some point yy above the ground givesabove the ground gives

½ ½ mvmvff22 + + mgymgy = = mghmgh

Conservation of Energy Conservation of Energy Example 2 (Pendulum)Example 2 (Pendulum)

• As the pendulum swings, As the pendulum swings, there is a continuous change there is a continuous change between potential and kinetic between potential and kinetic energiesenergies

• At At AA, the energy is potential, the energy is potential• At At BB, all of the potential , all of the potential

energy at energy at A A is transformed is transformed into kinetic energyinto kinetic energy– Let zero potential energy Let zero potential energy

be at be at BB• At At CC, the kinetic energy has , the kinetic energy has

been transformed back into been transformed back into potential energypotential energy

Conservation of Energy Conservation of Energy Example 3 (Spring Gun)Example 3 (Spring Gun)

• Choose point Choose point AA as the initial as the initial point and point and CC as the final point as the final point

EEAA = = EECC

KKAA + + UUgAgA + + UUsAsA = = KKCC + + UUgCgC + + UUsBsB

½ ½ kxkx2 2 = = mghmgh

WU Also valid for elastic potential energyAlso valid for elastic potential energy

Spring forceSpring force does does –W–W on block on block energy transfer from kinetic energy transfer from kinetic energy of the block to potential energy of the block to potential elastic energy of the spring.elastic energy of the spring.

Spring forceSpring force does does +W+W on block on block energy transfer from potential energy transfer from potential energy of the spring to kinetic energy of the spring to kinetic energy of the block.energy of the block.

fs

fs

Spring compressionSpring compression

Spring extensionSpring extension

III. Conservative / Nonconservative forcesIII. Conservative / Nonconservative forces

- If If WW11==WW22 always always conservative force. conservative force.

Examples:Examples: Gravitational force and spring force Gravitational force and spring force associated associated with potential energieswith potential energies..

- If If WW11≠≠WW22 nonconservative force. nonconservative force.

Examples:Examples: Drag force, frictional force Drag force, frictional force KEKE transferred into transferred into thermal energy. Non-reversible process.thermal energy. Non-reversible process.

- Thermal energy:- Thermal energy: Energy associated with the random Energy associated with the random movement of atoms and molecules. This is not a potential movement of atoms and molecules. This is not a potential energy.energy.

- Conservative force:Conservative force: The net work it does on a particle The net work it does on a particle moving around every closed path, from an initial point and then moving around every closed path, from an initial point and then back to that point is zero.back to that point is zero.

Conservative forceConservative force WWab,1ab,1== WWab,2ab,2

WWab,1ab,1+ W+ Wba,2ba,2=0 =0 W Wab,1ab,1= -W= -Wba,2ba,2

WWab,2ab,2= - W= - Wba,2ba,2

- The net work it does on a particle moving between two The net work it does on a particle moving between two points does not depend on the particle’s path.points does not depend on the particle’s path.

Proof:Proof:

WWab,2ab,2= W= Wab,1ab,1

IV. Determining potential energy valuesIV. Determining potential energy values

f

i

x

xUdxxFW )( Force Force FF is conservative is conservative

Gravitational potential energy:Gravitational potential energy:

Change in the gravitational potential energy Change in the gravitational potential energy of the particle-Earth system.of the particle-Earth system.

f

i

f

i

y

y ifyy ymgyymgymgdymgU )()(

mgyyUyU ii )(0,0

The gravitational potential energy associated with particle-The gravitational potential energy associated with particle-Earth system depends Earth system depends onlyonly on particle’s vertical position on particle’s vertical position “y”“y” relative to the reference position relative to the reference position y=0y=0, not on the horizontal , not on the horizontal position.position.

Reference configurationReference configuration

Elastic potential energy:Elastic potential energy:

Change in the elastic potential energy of the spring-block Change in the elastic potential energy of the spring-block system.system.

222

2

1

2

1

2)( i

x

x f

x

x kxkxxk

dxkxU f

i

f

i

2

2

1)(0,0 kxxUxU ii

Reference configuration Reference configuration when the spring is at its relaxed when the spring is at its relaxed length and the block is at xlength and the block is at xii=0.=0.

Remember!Remember! Potential energy is always associated with a Potential energy is always associated with a system.system.

EEmecmec= U + K= U + K

Assumptions:Assumptions: Only conservative forces cause energy transfer Only conservative forces cause energy transfer within the system.within the system.

UW

KW

11221212 0)()(0 UKUKUUKKUK

ΔΔEEmecmec= = ΔΔK + K + ΔΔU = 0U = 0

The system is isolated from its environment The system is isolated from its environment No external No external force from an object outside the system causes energy force from an object outside the system causes energy changes inside the system.changes inside the system.

V. Conservation of mechanical energyV. Conservation of mechanical energyMechanical energyMechanical energy of a system: Sum of the it’s potential (U) of a system: Sum of the it’s potential (U) and kinetic (K) energies.and kinetic (K) energies.

- In an In an isolated systemisolated system where where only conservative only conservative forcesforces cause energy changes, the kinetic energy and cause energy changes, the kinetic energy and potential energy can change, but their sum, the potential energy can change, but their sum, the mechanical energymechanical energy of the system of the system cannot changecannot change..

- When the When the mechanical energymechanical energy of a system is of a system is conservedconserved, we can relate the sum of kinetic energy , we can relate the sum of kinetic energy and potential energy at one instant to that at another and potential energy at one instant to that at another instant without considering the intermediate motion instant without considering the intermediate motion and without finding the work done by the forces and without finding the work done by the forces involved.involved.

A bead slides without friction around a loop-the-loop. The A bead slides without friction around a loop-the-loop. The bead is released from a height bead is released from a height hh = 3.50 = 3.50RR. . (a) What is (a) What is its speed at point its speed at point AA? (b) How large is the normal force on ? (b) How large is the normal force on it if its mass is it if its mass is 5.00 g5.00 g? ?

EEmecmec= = constantconstant

1122

0

UKUK

UKEmec

Potential energy curvesPotential energy curves

x

y

Finding the force analytically:Finding the force analytically:

)1()(

)()()( motionDdx

xdUxFxxFWxU

- The force is the negative of the slope of the curve The force is the negative of the slope of the curve U(x)U(x) versus versus xx..

- The particle’s kinetic energy is:The particle’s kinetic energy is: K(x) = EK(x) = Emecmec – U(x) – U(x)

VI. Work done on a system by an external forceVI. Work done on a system by an external force

No Friction:No Friction:

Work is energy transferWork is energy transfer “to”“to” or or “from”“from” a system by means of an a system by means of an external force acting on that system.external force acting on that system.

When more than one force acts on When more than one force acts on a system their net work is the energy a system their net work is the energy transferredtransferred toto oror fromfrom the system.the system.

W = W = ΔΔEEmecmec= = ΔΔK+ K+ ΔΔUU Ext. force Ext. force

dvvaadvv

mafF k

/)(5.02 20

220

2

Friction:Friction:

Remember!Remember! ΔΔEEmecmec= = ΔΔK+ K+ ΔΔU = 0U = 0 only when: only when:

- System isolated. - System isolated. - No external forces act on a system.No external forces act on a system.- All internal forces are conservative.All internal forces are conservative.

dfKFdW

dfmvmvFdvvd

mfF

k

kk

20

220

2

2

1

2

1)(

2

dfEFdW kmec

General:General: Example:Example: Block sliding up a ramp.Block sliding up a ramp.

dvvaadvv

mafF k

/)(5.02 20

220

2

A A 15.7 kg15.7 kg block is dragged over a rough, horizontal block is dragged over a rough, horizontal surface by asurface by a 72.2 N72.2 N force acting atforce acting at 21°21° above the horizontal. above the horizontal. The block is displacedThe block is displaced 4.5 m4.5 m, and the coefficient of kinetic , and the coefficient of kinetic friction isfriction is 0.1770.177. Find the work done on the block by (a) the. Find the work done on the block by (a) the 72.2 N72.2 N force, (b) the normal force, and (c) the gravitational force, (b) the normal force, and (c) the gravitational force. (d) What is the increase in internal energy of the block-force. (d) What is the increase in internal energy of the block-surface system due to friction? (e) Find the total change in the surface system due to friction? (e) Find the total change in the block's kinetic energy. block's kinetic energy.

21

F

d

y

F

Fx

Fy

N

Fgfk

A potential energy function for a two-dimensional A potential energy function for a two-dimensional force is of the form force is of the form U = 3xU = 3x33y – 7xy – 7x. Find the force . Find the force that act at the point (that act at the point (x,yx,y).).

dfE kth Thermal energy:Thermal energy:

thmec EEFdW

Work done on a system by an external force, friction involvedWork done on a system by an external force, friction involved

Friction due to cold welding Friction due to cold welding between two surfaces. As between two surfaces. As the block slides over the the block slides over the floor, the sliding causes floor, the sliding causes tearing and reforming of the tearing and reforming of the welds between the block and welds between the block and the floor, which makes the the floor, which makes the block-floor warmer.block-floor warmer.

VI. Conservation of energyVI. Conservation of energy

- The total energy of a system can only change by amounts The total energy of a system can only change by amounts of energy transferredof energy transferred “from”“from” or or “to”“to” the system.the system.

intEEEW thmec Experimental lawExperimental law

-The total energy of an isolated system cannot change. The total energy of an isolated system cannot change. (There cannot be energy transfers to or from it).(There cannot be energy transfers to or from it).

Total energy of a system = Total energy of a system = EEmechanicalmechanical + E + Ethermalthermal + E + Einternalinternal

Isolated system:Isolated system:

0int EEE thmec

In an isolated system we can relate the total energy at one In an isolated system we can relate the total energy at one instant to the total energy at another instant without instant to the total energy at another instant without considering the energies at intermediate states.considering the energies at intermediate states.

VII. External forces and internal energy changesVII. External forces and internal energy changes

Example:Example: skater pushes herself away from a railing. There is a skater pushes herself away from a railing. There is a forceforce FF on her from the railing that increases her kinetic energy.on her from the railing that increases her kinetic energy.

i)i) One part of an object (skater’s arm) One part of an object (skater’s arm) does not move like the rest of body.does not move like the rest of body.

ii) Internal energy transfer (from one part ii) Internal energy transfer (from one part of the system to another) via the of the system to another) via the external forceexternal force FF. . Biochemical energy Biochemical energy from muscles transferred to kinetic from muscles transferred to kinetic energy of the body.energy of the body.

cos

cos

)(cos

,

,

FdE

FdWUK

systemisolatedNon

dFKW

mec

extF

extF

Change in system’s mechanical energy by an external forceChange in system’s mechanical energy by an external force

cos

0

int

int

FdEE

EE

mec

mec

Change in system’s Change in system’s

internal energy by a internal energy by a external forceexternal force

Proof:Proof:

cos2

1

2

1

)(2

20

2

20

2

FdK

dMaMvMv

Mdavv

x

x

129. A massless rigid rod of length A massless rigid rod of length LL has a ball of mass has a ball of mass m m attached to one attached to one end. The other end is pivoted in such a way that the ball will move in a end. The other end is pivoted in such a way that the ball will move in a vertical circle. First, assume that there is no friction at the pivot. The vertical circle. First, assume that there is no friction at the pivot. The system is launched downward from the horizontal position system is launched downward from the horizontal position AA with initial with initial speed speed vv00.. The ball just barely reaches point The ball just barely reaches point DD and then stops. and then stops.

D

CA

B

L

v0

y

x

mg

TFc

(a)(a) Derive an expression forDerive an expression for vv00 in terms ofin terms of LL,, mm and and gg. .

(b) What is the tension in the rod when the ball passes (b) What is the tension in the rod when the ball passes throughthrough BB? ?

(c) A little girl is placed on the pivot to increase the (c) A little girl is placed on the pivot to increase the friction there. Then the ball just barely reachesfriction there. Then the ball just barely reaches CC when launched from when launched from A A with the same speed with the same speed as before. What is the decrease in mechanical as before. What is the decrease in mechanical energy during this motion?energy during this motion?

(d) What is the decrease in mechanical energy by (d) What is the decrease in mechanical energy by the time the ball finally comes to rest at the time the ball finally comes to rest at B B after after several oscillations?several oscillations?

7.7. A particle is attached between two identical springs on a horizontal A particle is attached between two identical springs on a horizontal frictionless table. Both springs have spring constant frictionless table. Both springs have spring constant kk and are initially and are initially unstressed. (a) If the particle is pulled a distance unstressed. (a) If the particle is pulled a distance xx along a direction along a direction perpendicular to the initial configuration of the springs show that the perpendicular to the initial configuration of the springs show that the force exerted by the springs on the particle is force exerted by the springs on the particle is

(b) Determine the amount of work done by this force in moving the (b) Determine the amount of work done by this force in moving the particle from particle from x = Ax = A to to x = x = 00; (c) show that the potential energy of the ; (c) show that the potential energy of the system is:system is:

F 2kx 1

L

x2 L2

i

U(x)kx2 2kLL x2 L2

61. In the figure below, a block slides along a path that is In the figure below, a block slides along a path that is without friction until the block reaches the section of length without friction until the block reaches the section of length L=0.75mL=0.75m, which begins at height , which begins at height h=2mh=2m. In that section, the . In that section, the coefficient of kinetic friction is coefficient of kinetic friction is 0.40.4. The block passes through . The block passes through point point A A with a speed of with a speed of

8m/s8m/s. Does it reach point . Does it reach point BB (where the section of (where the section of friction ends)? If so, what is friction ends)? If so, what is the speed there and if not, the speed there and if not, what greatest height above what greatest height above point point A A does it reach?does it reach?

mg

N

fC

101.101. A A 3kg 3kg sloth hangs sloth hangs 3m3m above the ground. (a) What is the above the ground. (a) What is the gravitational potential energy of the sloth-Earth system if we gravitational potential energy of the sloth-Earth system if we take the reference point take the reference point y=0 y=0 to be at the ground? If the sloth to be at the ground? If the sloth drops to the ground and air drag on it is assumed to be drops to the ground and air drag on it is assumed to be negligible, what are (b) the kinetic energy and (c) the speed of negligible, what are (b) the kinetic energy and (c) the speed of the sloth just before it reaches the ground?the sloth just before it reaches the ground?

130.130. A metal tool is sharpen by being held against the rim of A metal tool is sharpen by being held against the rim of a wheel on a grinding machine by a force of a wheel on a grinding machine by a force of 180N180N. The . The frictional forces between the rim and the tool grind small frictional forces between the rim and the tool grind small pieces of the tool. The wheel has a radius of pieces of the tool. The wheel has a radius of 20cm20cm and and rotates at rotates at 2.5 rev/s2.5 rev/s. The coefficient of kinetic friction between . The coefficient of kinetic friction between the wheel and the tool is the wheel and the tool is 0.320.32. At what rate is energy being . At what rate is energy being transferred from the motor driving the wheel and the tool to transferred from the motor driving the wheel and the tool to the kinetic energy of the material thrown from the tool?the kinetic energy of the material thrown from the tool?

F=180N

v

82.82. A block with a kinetic energy of A block with a kinetic energy of 30J30J is about to collide with a spring at its is about to collide with a spring at its relaxed length. As the block compresses the spring, a frictional force between relaxed length. As the block compresses the spring, a frictional force between the block and floor acts on the block. The figure below gives the kinetic energy the block and floor acts on the block. The figure below gives the kinetic energy of the block of the block (K(x))(K(x)) and the potential energy of the spring and the potential energy of the spring (U(x))(U(x)) as a function of as a function of the position the position x x of the block, as the spring is compressed. What is the of the block, as the spring is compressed. What is the increase in thermal energy of the block and the floor when (a) the block increase in thermal energy of the block and the floor when (a) the block reaches position reaches position 0.1 m0.1 m and (b) the spring reaches its maximum compression? and (b) the spring reaches its maximum compression?

f

mg

N