Advanced Problems 3 These problems will contain: Basic concepts of Work Usage of the potential and kinetic energy theorem The concept of the conservation of energy.

1. A 15kg block is dragged over a rough, horizontal surface by a 70N force acting 20° above the horizontal. The block is displaced 5m, and the coefficient of kinetic friction is .3. Find the work done by the 70N force The normal force The force of gravity What is the energy loss due to friction? Find the total change in the block’s kinetic energy.

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

3. The force acting on a particle is Fx=(8x-16)N, where x is in meters. Make a plot of this force verses x from x=0 to x=3m. From your graph, find the net work done by this force as the particle moves form x=0 to x=3m.

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

5. When a 4kg mass is hung vertically on a certain light spring that obeys Hook’s law, the spring stretches 2.5cm. If the 4kg mass is removed, How far will the spring stretch if a 1.5kg mass is hung on it and How much work must an external agent do to stretch the same spring 4cm from its unstretched position?

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

7. A small mass m is pulled to the top of a frictionless half cylinder. If the mass moves at a constant speed, show that F=mgcosθ. By directly integrating find the work done in moving the mass at constant speed from the bottom to the top of the half cylinder. Here ds represents an incremental displacement of the small mass. F m r θ

8. A 0.6kg particle has a speed of 2m/s at point A and kinetic energy of 7.5J at point B. What is its kinetic energy at A Its speed at B. The total work done on the particle as it moves from A to B.

9. A 3kg mass has an initial velocity vi=(6i-2j)m/s. What is its kinetic energy at this time? Find the total work done on the object if its velocity changes to (8i+4j)m/s. (hint: remember that v2=v●v

10. A mechanic pushes a 2500kg car, moving it from rest and making it accelerate from rest to a speed v. He does 5000J of work in the process. During this time, the car moves 25m. If friction between the car and the road is negligible, What is the final speed v of the car? What constant horizontal force did he exert on the car?

11. A 40kg box initially at rest is pushed 5m along a rough, horizontal floor with a constant applied horizontal force of 130N. If the coefficient of friction between the box and the floor is .3, find (a) The work done by the applied force. (b) The energy loss due to friction. (c) The work done by the normal force. (d) The work done by gravity. (e) The change in kinetic energy of the box. (f) The final speed of the box.

12. A block of mass 12kg slides from rest down a frictionless 35° incline and is stopped by a strong spring with k=3x104N/m. The block slides 3m from the point of release to the point where it comes to rest against the spring. When the block comes to rest, how far has the spring been compressed? What is the maximum compression of the spring?

13. A 650kg elevator starts from rest 13. A 650kg elevator starts from rest. It moves upward for 3 seconds with constant acceleration until it reaches its cruising speed of 1.75m/s. What is the average power of the elevator motor during this period? How does this power compare with its power when it moves at its cruising speed.

14. At 650-kg elevator starts from rest 14. At 650-kg elevator starts from rest. It moves upward for 3s with constant acceleration until it reaches its cruising speed of 1.75 m/s. What is the average power of the elevator motor during this period? How does this power compare with its power when it moves at its cruising speed?

(d)Is F conservative or nonconservative? Explain. 15. A force acting on a particle moving in the xy plane is given by F = (2yi + x2j)N, where x and y are in meters. The particle moves from the origin to a final position having coordinates x=5m and y=m, as in the figure. Calculate the work done by F along (a)OAC (b)OBC (c)OC (d)Is F conservative or nonconservative? Explain. y (5,5) m B C x O A

the vertical component of vi 16. A particle of mass 0.5kg is shot from P as shown in the figure. The particle has an initial velocity vi with a horizontal component of 30m/s. The particle rises to a maximum height of 20m above P. Using the law of conservation of energy, determine the vertical component of vi the work done by the gravitational force on the particle during its motion from P to B the horizontal and the vertical components of the velocity vector when the particle reaches B. 20m 60m

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

18. A 120g mass is attached to he bottom end of an unstressed spring 18. A 120g mass is attached to he bottom end of an unstressed spring. The spring is hanging vertically and has a spring constant of 40N/m. The mass has dropped. What is its maximum speed? How far does it drop before coming to rest momentarily?

(b)find the maximum height to which the 3kg mass rises. 19. Two masses are connected by a light string passing over a light frictionless pulley, as shown in the figure. The 5kg mass is released from rest. Using the law of conservation of energy, (a)determine the speed of the 3kg mass just as the 5kg mass hits the ground (b)find the maximum height to which the 3kg mass rises. m1=5kg h=4m m2=3kg

20. A 2kg ball is attached to the bottom end of a length of 10lb (44 20. A 2kg ball is attached to the bottom end of a length of 10lb (44.5N) fishing line. The top end of the fishing line is held stationary. The ball is released from rest while the held stationary. The ball is released from rest while the line is taut and horizontal (=90°). At what angle  (measured from the vertical) will the fishing line break?

21. After its release at the top of the first rise, a roller coaster car moves freely with negligible friction. The roller coaster 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, the riders are upside down and feel weightless. (a)find the speed of the roller coaster car at the top of the loop (position 3) (b)at position 1 (c)at position 2 (d)find the difference in height between positions 1 and 4 if the speed at position 4 is 10 m/s. 3 2 4 1

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