Advanced Problems 3Advanced Problems 3These problems will contain:These problems will contain:

1.1. Basic concepts of WorkBasic concepts of Work

2.2. Usage of the potential and Usage of the potential and kinetic energy theoremkinetic energy theorem

3.3. The concept of the The concept of the conservation of energy.conservation of energy.

1. A 15kg block is dragged over a rough, 1. A 15kg block is dragged over a rough, horizontal surface by a 70N force horizontal surface by a 70N force acting 20° above the horizontal. The acting 20° above the horizontal. The block is displaced 5m, and the block is displaced 5m, and the coefficient of kinetic friction is .3. coefficient of kinetic friction is .3. Find the work done by theFind the work done by the

(a)(a)70N force70N force

(b)(b)The normal forceThe normal force

(c)(c) The force of gravityThe force of gravity

(d)(d)What is the energy loss due to friction?What is the energy loss due to friction?

(e)(e)Find the total change in the block’s Find the total change in the block’s kinetic energy.kinetic energy.

2. Vector A extends from the 2. Vector A extends from the origin to a point having polar origin to a point having polar coordinates (7,70°), and vector B coordinates (7,70°), and vector B extends form the origin to a extends form the origin to a point having polar coordinates point having polar coordinates (4,130°). Find A●B.(4,130°). Find A●B.

3. The force acting on a particle is 3. The force acting on a particle is

FFxx=(8x-16)N, where x is in meters.=(8x-16)N, where x is in meters.

(a)(a) Make a plot of this force verses x from x=0 to Make a plot of this force verses x from x=0 to x=3m.x=3m.

(b)(b) From your graph, find the net work done by From your graph, find the net work done by this force as the particle moves form x=0 to this force as the particle moves form x=0 to x=3m.x=3m.

4. A force F=(4x4. A force F=(4xii + 3y + 3yjj)N acts on an )N acts on an object as it moves in the x direction object as it moves in the x direction from the origin to x=5m. Find the from the origin to x=5m. Find the work work done on the done on the object by the force.object by the force.

drFW

5. When a 4kg mass is hung vertically 5. When a 4kg mass is hung vertically on a certain light spring that obeys on a certain light spring that obeys Hook’s law, the spring stretches Hook’s law, the spring stretches 2.5cm. If the 4kg mass is removed, 2.5cm. If the 4kg mass is removed,

(a)(a)How far will the spring stretch if a How far will the spring stretch if a 1.5kg mass is hung on it and1.5kg mass is hung on it and

(b)(b)How much work must an external How much work must an external agent do to stretch the same spring agent do to stretch the same spring 4cm from its unstretched position?4cm from its unstretched position?

6. If it takes 4J of work to stretch a 6. If it takes 4J of work to stretch a Hook’s law spring 10cm from its Hook’s law spring 10cm from its unstressed length, determine the unstressed length, determine the extra work required to stretch it an extra work required to stretch it an additional 10cm.additional 10cm.

7. A small mass 7. A small mass mm is pulled to the top of a is pulled to the top of a frictionless half cylinder.frictionless half cylinder.

(a)(a) If the mass moves at a constant speed, If the mass moves at a constant speed, show that show that F=mgcosF=mgcosθθ..

(b)(b) By directly integrating By directly integrating find find the work done in moving the mass at the work done in moving the mass at constant speed from the bottom to the constant speed from the bottom to the top of the half cylinder. Here top of the half cylinder. Here ddss represents an incremental displacement represents an incremental displacement of the small mass.of the small mass.

dsFW

F

m

rθ

8. A 0.6kg particle has a speed of 2m/s at 8. A 0.6kg particle has a speed of 2m/s at point A and kinetic energy of 7.5J at point point A and kinetic energy of 7.5J at point B.B.

(a)(a) What is its kinetic energy at AWhat is its kinetic energy at A

(b)(b) Its speed at B.Its speed at B.

(c)(c) The total work done on the particle as it The total work done on the particle as it moves from A to B.moves from A to B.

9. A 3kg mass has an initial velocity v9. A 3kg mass has an initial velocity vii=(6=(6ii--22jj)m/s. )m/s.

(a)(a) What is its kinetic energy at this time?What is its kinetic energy at this time?

(b)(b) Find the total work done on the object if Find the total work done on the object if its velocity changes to (8its velocity changes to (8ii+4+4jj)m/s. (hint: )m/s. (hint: remember that vremember that v22=v●v=v●v

10. A mechanic pushes a 2500kg car, moving 10. A mechanic pushes a 2500kg car, moving it from rest and making it accelerate from it from rest and making it accelerate from rest to a speed rest to a speed vv. He does 5000J of work . He does 5000J of work in the process. During this time, the car in the process. During this time, the car moves 25m. If friction between the car moves 25m. If friction between the car and the road is negligible, and the road is negligible,

(a)(a) What is the final speed What is the final speed vv of the car? of the car?

(b)(b) What constant horizontal force did he What constant horizontal force did he exert on the car?exert on the car?

11. A 40kg box initially at rest is pushed 5m 11. A 40kg box initially at rest is pushed 5m along a rough, horizontal floor with a along a rough, horizontal floor with a constant applied horizontal force of constant applied horizontal force of 130N. If the coefficient of friction 130N. If the coefficient of friction between the box and the floor is .3, findbetween the box and the floor is .3, find

(a) The work done by the applied force.(a) The work done by the applied force.

(b) The energy loss due to friction.(b) The energy loss due to friction.

(c) The work done by the normal force.(c) The work done by the normal force.

(d) The work done by gravity.(d) The work done by gravity.

(e) The change in kinetic energy of the box.(e) The change in kinetic energy of the box.

(f) The final speed of the box.(f) The final speed of the box.

12. A block of mass 12kg slides from 12. A block of mass 12kg slides from rest down a frictionless 35° incline rest down a frictionless 35° incline and is stopped by a strong spring and is stopped by a strong spring with k=3x10with k=3x1044N/m. The block slides N/m. The block slides 3m from the point of release to the 3m from the point of release to the point where it comes to rest point where it comes to rest against the spring. against the spring.

(a)(a)When the block comes to rest, how When the block comes to rest, how far has the spring been far has the spring been compressed?compressed?

(b)(b)What is the maximum compression What is the maximum compression of the spring?of the spring?

13. A 650kg elevator starts from rest. 13. A 650kg elevator starts from rest. It moves upward for 3 seconds with It moves upward for 3 seconds with constant acceleration until it constant acceleration until it reaches its cruising speed of reaches its cruising speed of 1.75m/s.1.75m/s.

(a)(a)What is the average power of the What is the average power of the elevator motor during this period?elevator motor during this period?

(b)(b)How does this power compare with How does this power compare with its power when it moves at its its power when it moves at its cruising speed. cruising speed.

14. At 650-kg elevator starts from 14. At 650-kg elevator starts from rest. It moves upward for 3s with rest. It moves upward for 3s with constant acceleration until it constant acceleration until it reaches its cruising speed of 1.75 reaches its cruising speed of 1.75 m/s. m/s.

(a)(a)What is the average power of the What is the average power of the elevator motor during this period?elevator motor during this period?

(b)(b)How does this power compare with How does this power compare with its power when it moves at its its power when it moves at its cruising speed?cruising speed?

15. A force acting on a particle moving in the xy 15. A force acting on a particle moving in the xy plane is given by F = (2yi + xplane is given by F = (2yi + x22j)N, where x and y j)N, where x and y are in meters. The particle moves from the origin are in meters. The particle moves from the origin to a final position having coordinates x=5m and to a final position having coordinates x=5m and y=m, as in the figure. Calculate the work done by y=m, as in the figure. Calculate the work done by F alongF along

(a)OAC(a)OAC

(b)OBC(b)OBC

(c)OC(c)OC

(d)Is F conservative or nonconservative? Explain.(d)Is F conservative or nonconservative? Explain.

y

xO

B C

A

(5,5) m

16. A particle of mass 0.5kg is shot from P as 16. A particle of mass 0.5kg is shot from P as shown in the figure. The particle has an initial shown in the figure. The particle has an initial velocity vvelocity vii with a horizontal component of with a horizontal component of 30m/s. The particle rises to a maximum height 30m/s. The particle rises to a maximum height of 20m above P. Using the law of conservation of 20m above P. Using the law of conservation of energy, determineof energy, determine

(a)(a) the vertical component of vthe vertical component of vii

(b)(b) the work done by the gravitational force on the the work done by the gravitational force on the particle during its motion from P to Bparticle during its motion from P to B

(c)(c) the horizontal and the vertical components of the horizontal and the vertical components of the velocity vector when the particle reaches B.the velocity vector when the particle reaches B.

60m

20m

17. A bead slides without friction around a 17. A bead slides without friction around a loop-the-loop. If the bead is released loop-the-loop. If the bead is released from a height h=3.5R, what is its speed from a height h=3.5R, what is its speed at point A? How great is the normal force at point A? How great is the normal force on it if its mass is 5g?on it if its mass is 5g?

h

A

18. A 120g mass is attached to he 18. A 120g mass is attached to he bottom end of an unstressed bottom end of an unstressed spring. The spring is hanging spring. The spring is hanging vertically and has a spring constant vertically and has a spring constant of 40N/m. The mass has dropped.of 40N/m. The mass has dropped.

(a)(a)What is its maximum speed?What is its maximum speed?

(b)(b)How far does it drop before coming How far does it drop before coming to rest momentarily?to rest momentarily?

19. Two masses are connected by a light 19. Two masses are connected by a light string passing over a light frictionless string passing over a light frictionless pulley, as shown in the figure. The 5kg pulley, as shown in the figure. The 5kg mass is released from rest. Using the law mass is released from rest. Using the law of conservation of energy,of conservation of energy,

(a)determine the speed of the 3kg mass just (a)determine the speed of the 3kg mass just as the 5kg mass hits the groundas the 5kg mass hits the ground

(b)find the maximum height to which the (b)find the maximum height to which the 3kg mass rises.3kg mass rises.

m1=5kg

m2=3kg h=4m

20. A 2kg ball is attached to the 20. A 2kg ball is attached to the bottom end of a length of 10lb bottom end of a length of 10lb (44.5N) fishing line. The top end of (44.5N) fishing line. The top end of the fishing line is held stationary. the fishing line is held stationary. The ball is released from rest while The ball is released from rest while the held stationary. The ball is the held stationary. The ball is released from rest while the line is released from rest while the line is taut and horizontal (taut and horizontal (=90°). At =90°). At what angle what angle (measured from the (measured from the vertical) will the fishing line break?vertical) will the fishing line break?

21. After its release at the top of the first rise, a roller coaster 21. After its release at the top of the first rise, a roller coaster car moves freely with negligible friction. The roller coaster car moves freely with negligible friction. The roller coaster shown in the figure, has a circular loop of radius of 20m. The shown in the figure, has a circular loop of radius of 20m. The car barely makes it around the loop: At the top of the loop, car barely makes it around the loop: At the top of the loop, the riders are upside down and feel weightless.the riders are upside down and feel weightless.

(a)find the speed of the roller coaster car at the top of the loop (a)find the speed of the roller coaster car at the top of the loop (position 3)(position 3)

(b)at position 1(b)at position 1

(c)at position 2(c)at position 2

(d)find the difference in height between positions 1 and 4 if the (d)find the difference in height between positions 1 and 4 if the speed at position 4 is 10 m/s.speed at position 4 is 10 m/s.

1

2

3

4

22. A 10kg block is released from point A. The track 22. A 10kg block is released from point A. The track is frictionless except for the portion between B and is frictionless except for the portion between B and C, which has a length of 6m. The block travels C, which has a length of 6m. The block travels down the track, hits a spring force of constant down the track, hits a spring force of constant k=2250N/m, and compresses the spring 0.3m from k=2250N/m, and compresses the spring 0.3m from its equilibrium position before coming to rest its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic momentarily. Determine the coefficient of kinetic friction between the block and the rough surface friction between the block and the rough surface between B and C.between B and C.

A

B C