Physics 4A Fall 2017 Final Exam

Name:

Dec 12, 2017

Please show your work! Answers are not complete without clear reasoning. When askedfor an expression, you must give your answer in terms of the variables given in the questionand/or fundamental constants.

Answer as many questions as you can, in any order. Do not forget to include appropriateunits when giving a number as an answer. Calculators are allowed. Notes, books, andinternet-connectable devices are not allowed. If you detach any pages from the test, pleasewrite your name on every page.

Constants

G = 6.67 × 10−11 N m2 kg−2

g = 9.8 m s−2

1

1. A particle starts from rest and accelerates as shown, moving in the x-direction.

(a) During which time interval(s) is the speed of the particle constant? [1 pt]

(b) Determine the particle’s speed at t = 5.0 s, t = 10.0 s, and at t = 20.0 s. [6 pts]

(c) Determine the distance traveled in the 20.0 s shown. [7 pts]

52 Chapter 2 Motion in One Dimension

horizontal axis or on the velocity or acceleration axes, but show the correct graph shapes.

vS

20 cm

40 cm 60 cm

100 cm

0

Figure P2.16

17. Figure P2.17 shows a graph of vx versus t for the motion of a motorcyclist as he starts from rest and moves along the road in a straight line. (a) Find the average accel-eration for the time interval t 5 0 to t 5 6.00 s. (b) Esti-mate the time at which the acceleration has its greatest positive value and the value of the acceleration at that instant. (c) When is the acceleration zero? (d) Estimate the maximum negative value of the acceleration and the time at which it occurs.

0 2 4 6 108 12t (s)

2468

10vx (m/s)

Figure P2.17

18. (a) Use the data in Problem 5 to construct a smooth graph of position versus time. (b) By constructing tan-gents to the x(t) curve, find the instantaneous velocity of the car at several instants. (c) Plot the instantaneous velocity versus time and, from this information, deter-mine the average acceleration of the car. (d) What was the initial velocity of the car?

19. A particle starts from rest and accelerates as shown in Figure P2.19. Deter-mine (a)  the particle’s speed at t 5 10.0 s and at t 5 20.0 s, and (b) the dis-tance traveled in the first 20.0 s.

20. An object moves along the x axis according to the equation x 5 3.00t 2 2 2.00t 1 3.00, where x is in meters and t is in seconds. Determine (a) the average speed between t 5 2.00 s and t 5 3.00 s, (b) the instan-taneous speed at t 5 2.00 s and at t 5 3.00 s, (c) the average acceleration between t 5 2.00 s and t 5 3.00 s, and (d) the instantaneous acceleration at t 5 2.00 s and t 5 3.00 s. (e) At what time is the object at rest?

21. A particle moves along the x axis according to the equation x 5 2.00 1 3.00t 2 1.00t 2, where x is in meters and t is in seconds. At t 5 3.00 s, find (a) the position of the particle, (b) its velocity, and (c) its acceleration.

2ax (m/s2)

0

1

!3

!2

5 10 15 20t (s)

!1

Figure P2.19

W

W

M

their respective speeds. (a) How far is the tortoise from the finish line when the hare resumes the race? (b) For how long in time was the hare stationary?

12. A car travels along a straight line at a constant speed of 60.0 mi/h for a distance d and then another distance d in the same direction at another constant speed. The average velocity for the entire trip is 30.0 mi/h. (a) What is the constant speed with which the car moved during the second distance d ? (b) What If? Suppose the second distance d were traveled in the opposite direction; you forgot something and had to return home at the same constant speed as found in part (a). What is the average velocity for this trip? (c) What is the average speed for this new trip?

13. A person takes a trip, driving with a constant speed of 89.5 km/h, except for a 22.0-min rest stop. If the per-son’s average speed is 77.8 km/h, (a) how much time is spent on the trip and (b) how far does the person travel?

Section 2.4 Acceleration 14. Review. A 50.0-g Super Ball traveling at 25.0 m/s bounces

off a brick wall and rebounds at 22.0 m/s. A high-speed camera records this event. If the ball is in contact with the wall for 3.50 ms, what is the magnitude of the aver-age acceleration of the ball during this time interval?

15. A velocity–time graph for an object moving along the x axis is shown in Figure P2.15. (a) Plot a graph of the acceleration versus time. Determine the average accel-eration of the object (b) in the time interval t 5 5.00 s to t 5 15.0 s and (c) in the time interval t 5 0 to t 5 20.0 s.

5t (s)

68

20

4

–4–2

–8–6

vx (m/s)

15 2010

Figure P2.15

16. A child rolls a marble on a bent track that is 100 cm long as shown in Figure P2.16. We use x to represent the position of the marble along the track. On the hor-izontal sections from x 5 0 to x 5 20 cm and from x 5 40 cm to x 5 60  cm, the marble rolls with constant speed. On the sloping sections, the marble’s speed changes steadily. At the places where the slope changes, the marble stays on the track and does not undergo any sudden changes in speed. The child gives the marble some initial speed at x 5 0 and t 5 0 and then watches it roll to x 5 90 cm, where it turns around, eventually returning to x 5 0 with the same speed with which the child released it. Prepare graphs of x versus t, vx versus t, and ax versus t, vertically aligned with their time axes identical, to show the motion of the marble. You will not be able to place numbers other than zero on the

AMT

M

W

2

2. A projectile is fired in such a way that its horizontal range (landing at the same heightit is launched from) is equal to twice its maximum height.

(a) What is the angle of projection? [7 pts]

(b) What is the ratio of the x-component of the initial velocity to the initial speed? [3 pts]

For both parts, your answer should be a number, not an expression.

3

3. An Atwood machine is placed in an elevator that accelerates downward with an ac-celeration of magnitude ae. You may assume m2 > m1 and the pulley is massless andfrictionless.

(a) What is the acceleration of m1 as seen by an observer in the elevator? [7 pts]

(b) What is the acceleration of m1 as seen by an observer on the the ground outsidethe elevator? [2 pts]

(c) What is the tension in the rope connecting the masses? [3 pts]

You must show an argument that considers the forces on the masses to get full credit.(You cannot just write down the answers.)

5.7 Analysis Models Using Newton’s Second Law 127

Conceptualize The reading on the scale is related to the extension of the spring in the scale, which is related to the force on the end of the spring as in Figure 5.2. Imagine that the fish is hanging on a string attached to the end of the spring. In this case, the magnitude of the force exerted on the spring is equal to the tension T in the string. There-fore, we are looking for T. The force T

S pulls down on the

string and pulls up on the fish.

Categorize We can categorize this problem by identify-ing the fish as a particle in equilibrium if the elevator is not accelerating or as a particle under a net force if the elevator is accelerating.

Analyze Inspect the diagrams of the forces acting on the fish in Figure 5.13 and notice that the external forces acting on the fish are the downward gravitational force F

Sg 5 mgS

and the force TS

exerted by the string. If the elevator is either at rest or moving at constant velocity, the fish is a par-ticle in equilibrium, so o Fy 5 T 2 Fg 5 0 or T 5 Fg 5 mg. (Remember that the scalar mg is the weight of the fish.) Now suppose the elevator is moving with an acceleration aS relative to an observer standing outside the elevator in an inertial frame. The fish is now a particle under a net force.

S O L U T I O N

123

45678

9 0TS

mgS

123

45678

9 0

TS

mgS

a b

aSaS

When the elevator accelerates upward, the spring scale reads a value greater than theweight of the fish.

When the elevator accelerates downward, the spring scale reads a value less than theweight of the fish.

Figure 5.13 (Example 5.8) A fish is weighed on a spring scale in an accelerating elevator car.

Apply Newton’s second law to the fish: o Fy 5 T 2 mg 5 may

Solve for T : (1) T 5 may 1 mg 5 mg aay

g1 1b 5 Fg aay

g1 1b

where we have chosen upward as the positive y direction. We conclude from Equation (1) that the scale reading T is greater than the fish’s weight mg if aS is upward, so ay is positive (Fig. 5.13a), and that the reading is less than mg if aS is downward, so ay is negative (Fig. 5.13b).

(B) Evaluate the scale readings for a 40.0-N fish if the elevator moves with an acceleration ay 5 62.00 m/s2.

S O L U T I O N

Evaluate the scale reading from Equation (1) if aS is upward:

T 5 140.0 N 2 a2.00 m/s2

9.80 m/s2 1 1b 5 48.2 N

Evaluate the scale reading from Equation (1) if aS is downward:

T 5 140.0 N 2 a22.00 m/s2

9.80 m/s2 1 1b 5 31.8 N

Finalize Take this advice: if you buy a fish in an elevator, make sure the fish is weighed while the elevator is either at rest or accelerating downward! Furthermore, notice that from the information given here, one cannot determine the direction of the velocity of the elevator.

Suppose the elevator cable breaks and the elevator and its contents are in free fall. What happens to the reading on the scale?

Answer If the elevator falls freely, the fish’s acceleration is ay 5 2g. We see from Equation (1) that the scale reading T is zero in this case; that is, the fish appears to be weightless.

WHAT IF ?

▸ 5.8 c o n t i n u e d

e

128 Chapter 5 The Laws of Motion

Example 5.9 The Atwood Machine

When two objects of unequal mass are hung vertically over a frictionless pulley of negligible mass as in Figure 5.14a, the arrangement is called an Atwood machine. The device is sometimes used in the laboratory to determine the value of g. Deter-mine the magnitude of the acceleration of the two objects and the tension in the lightweight string.

Conceptualize Imagine the situation pictured in Figure 5.14a in action: as one object moves upward, the other object moves downward. Because the objects are connected by an inextensible string, their accelerations must be of equal magnitude.

Categorize The objects in the Atwood machine are subject to the gravitational force as well as to the forces exerted by the strings connected to them. Therefore, we can categorize this problem as one involving two particles under a net force.

Analyze The free-body diagrams for the two objects are shown in Figure 5.14b. Two forces act on each object: the upward force T

S exerted by the string and

the downward gravitational force. In problems such as this one in which the pulley is modeled as massless and frictionless, the tension in the string on both sides of the pulley is the same. If the pulley has mass or is subject to friction, the tensions on either side are not the same and the situation requires techniques we will learn in Chapter 10. We must be very careful with signs in problems such as this one. In Figure 5.14a, notice that if object 1 accelerates upward, object 2 accelerates downward. Therefore, for consistency with signs, if we define the upward direction as positive for object 1, we must define the downward direction as positive for object 2. With this sign convention, both objects accelerate in the same direction as defined by the choice of sign. Furthermore, according to this sign conven-tion, the y component of the net force exerted on object 1 is T 2 m1g, and the y component of the net force exerted on object 2 is m2g 2 T.

AM

S O L U T I O N

Figure 5.14 (Example 5.9) The Atwood machine. (a) Two objects connected by a massless inextensible string over a frictionless pulley. (b) The free-body diagrams for the two objects.

+

+

m1

m1

m2

m2

a b

TS

TS

gS

gS

m1m2

From the particle under a net force model, apply New-ton’s second law to object 1:

(1) o Fy 5 T 2 m1g 5 m1ay

Apply Newton’s second law to object 2: (2) o Fy 5 m2g 2 T 5 m2ay

Add Equation (2) to Equation (1), noticing that T cancels: 2 m1g 1 m2g 5 m1ay 1 m2ay

Solve for the acceleration: (3) ay 5 am2 2 m1

m1 1 m2bg

Substitute Equation (3) into Equation (1) to find T : (4) T 5 m1(g 1 ay) 5 a 2m1m2

m1 1 m2bg

Finalize The acceleration given by Equation (3) can be interpreted as the ratio of the magnitude of the unbalanced force on the system (m2 2 m1)g to the total mass of the system (m1 1 m2), as expected from Newton’s second law. Notice that the sign of the acceleration depends on the relative masses of the two objects.

Describe the motion of the system if the objects have equal masses, that is, m1 5 m2.

Answer If we have the same mass on both sides, the system is balanced and should not accelerate. Mathematically, we see that if m1 5 m2, Equation (3) gives us ay 5 0.

What if one of the masses is much larger than the other: m1 .. m2?

Answer In the case in which one mass is infinitely larger than the other, we can ignore the effect of the smaller mass. Therefore, the larger mass should simply fall as if the smaller mass were not there. We see that if m1 .. m2, Equation (3) gives us ay 5 2g.

WHAT IF ?

WHAT IF ?

4

4. A block of mass m is attached to a spring of force constant k as shown. The block ispulled to a position xi to the right of equilibrium and released from rest at time t = 0.

If the surface is frictionless:

(a) Write down an expression for the equation of motion of the block (that is, specifyhow its position depends on time). If your expression uses other variables notgiven in this question, you must give their values. (You do not have to derive thisexpression.) [4 pts]

(b) What is the time period for this motion? [1 pt]

(c) Find the speed the block has as it passes through equilibrium. [4 pts]

If the surface has friction, and the speed that the block has when it first passes itsequilibrium position is one-quarter the value found in part (c):

(d) Find an expression for the coefficient of kinetic friction µk. [5 pts]

Problems 237

duced high-frequency “microtremor” vibrations that were rapidly damped and did not travel far. Assume 0.01% of the total energy was carried away by long-range seismic waves. The magnitude of an earthquake on the Richter scale is given by

M 5log E 2 4.8

1.5 where E is the seismic wave energy in joules. According

to this model, what was the magnitude of the demon-stration quake?

11. Review. The system shown in Figure P8.11 consists of a light, inextensible cord, light, frictionless pulleys, and blocks of equal mass. Notice that block B is attached to one of the pul-leys. The system is initially held at rest so that the blocks are at the same height above the ground. The blocks are then released. Find the speed of block A at the moment the vertical separation of the blocks is h.

Section 8.3 Situations Involving Kinetic Friction 12. A sled of mass m is given a kick on a frozen pond. The

kick imparts to the sled an initial speed of 2.00 m/s. The coefficient of kinetic friction between sled and ice is 0.100. Use energy considerations to find the distance the sled moves before it stops.

13. A sled of mass m is given a kick on a frozen pond. The kick imparts to the sled an initial speed of v. The coef-ficient of kinetic friction between sled and ice is mk. Use energy considerations to find the distance the sled moves before it stops.

14. A crate of mass 10.0 kg is pulled up a rough incline with an initial speed of 1.50 m/s. The pulling force is 100 N parallel to the incline, which makes an angle of 20.08 with the horizontal. The coefficient of kinetic friction is 0.400, and the crate is pulled 5.00 m. (a) How much work is done by the gravitational force on the crate? (b) Determine the increase in internal energy of the crate–incline system owing to friction. (c) How much work is done by the 100-N force on the crate? (d) What is the change in kinetic energy of the crate? (e) What is the speed of the crate after being pulled 5.00 m?

15. A block of mass m 5 2.00 kg is attached to a spring of force constant k 5 500 N/m as shown in Figure P8.15. The block is pulled to a posi-tion xi 5 5.00 cm to the right of equilibrium and released from rest. Find the speed the block has as it passes through equilibrium if (a) the horizontal surface is frictionless and (b) the coefficient of friction between block and surface is mk 5 0.350.

16. A 40.0-kg box initially at rest is pushed 5.00 m along a rough, horizontal floor with a constant applied horizontal force of 130 N. The coefficient of friction

BA

Figure P8.11

S

S

M

x ! xi

km

x ! 0

Figure P8.15

W

is its speed at point !? (b) How large is the normal force on the bead at point ! if its mass is 5.00 g?

6. A block of mass m 5 5.00 kg is released from point ! and slides on the frictionless track shown in Figure P8.6. Determine (a) the block’s speed at points " and # and (b) the net work done by the gravitational force on the block as it moves from point ! to point #.

2.00 m

5.00 m3.20 m

m!

"

#

Figure P8.6

7. Two objects are connected by a light string passing over a light, frictionless pulley as shown in Figure P8.7. The object of mass m1 5 5.00 kg is released from rest at a height h 5 4.00 m above the table. Using the isolated sys-tem model, (a) determine the speed of the object of mass m2 5 3.00  kg just as the 5.00-kg object hits the table and (b) find the maxi-mum height above the table to which the 3.00-kg object rises.

8. Two objects are connected by a light string passing over a light, frictionless pulley as shown in Figure P8.7. The object of mass m1 is released from rest at height h above the table. Using the isolated system model, (a) determine the speed of m2 just as m1 hits the table and (b) find the maximum height above the table to which m2 rises.

9. A light, rigid rod is 77.0 cm long. Its top end is piv-oted on a frictionless, horizontal axle. The rod hangs straight down at rest with a small, massive ball attached to its bottom end. You strike the ball, suddenly giving it a horizontal velocity so that it swings around in a full circle. What minimum speed at the bottom is required to make the ball go over the top of the circle?

10. At 11:00 a.m. on September 7, 2001, more than one million British schoolchildren jumped up and down for one minute to simulate an earthquake. (a) Find the energy stored in the children’s bodies that was con-verted into internal energy in the ground and their bodies and propagated into the ground by seismic waves during the experiment. Assume 1 050 000 chil-dren of average mass 36.0 kg jumped 12 times each, raising their centers of mass by 25.0 cm each time and briefly resting between one jump and the next. (b) Of the energy that propagated into the ground, most pro-

W

h

m1

m2

Figure P8.7 Problems 7 and 8.

M

S

5

5. A light string is wound around a uniform disk of radius R and mass M . The disk isreleased from rest with the string vertical and its top end tied to a fixed bar.

(a) Show that the tension in the string is one third of the weight of the disk. [7 pts]

(b) Find the magnitude of the acceleration of the center of mass of the disk. [2 pts]

(c) Find the speed of the center of mass after the disk has descended through distanceh. [3 pts]

332 Chapter 10 Rotation of a Rigid Object About a Fixed Axis

that breaks loose from the tire on one turn rises h 5 54.0 cm above the tangent point. A drop that breaks loose on the next turn rises 51.0 cm above the tangent point. The height to which the drops rise decreases because the angular speed of the wheel decreases. From this information, determine the magnitude of the average angular acceleration of the wheel.

h

v ! 0

Figure P10.74 Problems 74 and 75.

75. A bicycle is turned upside down while its owner repairs a flat tire on the rear wheel. A friend spins the front wheel, of radius R, and observes that drops of water fly off tangentially in an upward direction when the drops are at the same level as the center of the wheel. She measures the height reached by drops moving ver-tically (Fig. P10.74). A drop that breaks loose from the tire on one turn rises a distance h1 above the tangent point. A drop that breaks loose on the next turn rises a distance h2 , h1 above the tangent point. The height to which the drops rise decreases because the angular speed of the wheel decreases. From this information, determine the magnitude of the average angular accel-eration of the wheel.

76. (a) What is the rotational kinetic energy of the Earth about its spin axis? Model the Earth as a uniform sphere and use data from the endpapers of this book. (b) The rotational kinetic energy of the Earth is decreasing steadily because of tidal friction. Assuming the rotational period decreases by 10.0 ms each year, find the change in one day.

77. Review. As shown in Figure P10.77, two blocks are con-nected by a string of negligible mass passing over a pul-ley of radius r = 0.250 m and moment of inertia I. The block on the frictionless incline is moving with a con-stant acceleration of magnitude a = 2.00  m/s2. From this information, we wish to find the moment of inertia of the pulley. (a)  What analysis model is appropriate for the blocks? (b) What analysis model is appropriate

S

GP

for the pulley? (c) From the analysis model in part (a), find the tension T 1. (d) Similarly, find the tension T 2. (e) From the analysis model in part (b), find a symbolic expression for the moment of inertia of the pulley in terms of the tensions T1 and T2, the pulley radius r, and the acceleration a. (f) Find the numerical value of the moment of inertia of the pulley.

78. Review. A string is wound around a uniform disk of radius R and mass M. The disk is released from rest with the string vertical and its top end tied to a fixed bar (Fig. P10.78). Show that (a)  the tension in the string is one third of the weight of the disk, (b) the magnitude of the acceleration of the center of mass is 2g/3, and (c) the speed of the cen-ter of mass is (4gh/3)1/2 after the disk has descended through distance h. (d) Verify your answer to part (c) using the energy approach.

79. The reel shown in Figure P10.79 has radius R and moment of inertia I. One end of the block of mass m is connected to a spring of force constant k, and the other end is fastened to a cord wrapped around the reel. The reel axle and the incline are frictionless. The reel is wound counterclockwise so that the spring stretches a distance d from its unstretched position and the reel is then released from rest. Find the angular speed of the reel when the spring is again unstretched.

RI

k

u

m

Figure P10.79

80. A common demonstration, illustrated in Figure P10.80, consists of a ball resting at one end of a uniform board of length , that is hinged at the other end and elevated at an angle u. A light cup is attached to the board at rc so that it will catch the ball when the support stick is removed suddenly. (a) Show that the ball will lag behind the falling board when u is less than 35.38.

h

MR

Figure P10.78

S

S

37.0"

15.0 kg

20.0 kg

T2T1

m2m1

aS

Figure P10.77

Cup

!

Hinged end

Supportstick

u

rc

Figure P10.80

6

6. A small lump of putty, of mass m, moving initially with velocity v, strikes the bottomoutside edge of a uniform circular disk of mass M = 8m and radius R that hangsvertically downward from a frictionless pivot at the point O on its top edge as shown.The initial velocity v is perpendicular to a line drawn downward from O through thecenter of mass of the disk. The putty sticks to the disk after striking it.

m v

CM

O

Point where putty sticksIncoming putty

g

Without calculating:

(a) Is linear momentum conserved in the collision? Why or why not? [2 pts]

(b) Is angular momentum conserved about axis O in the collision? Why or whynot? [2 pts]

(c) Is kinetic energy conserved in the collision? Why or why not? [2 pts]

After being struck the disk swings to the right.

(d) To what height above its initial position does the center of mass of the disk risebefore coming momentarily to rest? [11 pts]

7

7. A uniform beam of mass m is inclined at an angle θ to the horizontal. Its upper end(point P) produces a 90◦ bend in a very rough rope tied to a wall, and its lower endrests on a rough floor. Let µs represent the coefficient of static friction between beamand floor. In terms of m, M , and µs:

(a) find θmin the minimum value the angle θ can take if the beam is not to slip. [9 pts]

(b) find the magnitude of the reaction force on the beam at the floor, given θ =θmin. [2 pts]

Problems 385

shown in Figure P12.50a. The rope make an angle u 5 37.08 with the floor and is tied h1 5 10.0 cm from the bottom of the cabinet. The uniform rectangular cabi-net has height , 5 100 cm and width w 5 60.0 cm, and it weighs 400 N. The cabinet slides with constant speed when a force F 5 300 N is applied through the rope. The worker tires of walking backward. He fastens the rope to a point on the cabinet h2  5 65.0 cm off the floor and lays the rope over his shoulder so that he can walk forward and pull as shown in Figure P12.50b. In this way, the rope again makes an angle of u 5 37.08 with the horizontal and again has a tension of 300 N. Using this technique, the worker is able to slide the cabinet over a long distance on the floor without tiring.

a

FS

b

h2

uFSw

!

h1

u

w

!

Figure P12.50 Problems 50 and 62.

51. A uniform beam of mass m is inclined at an angle u to the horizontal. Its upper end (point P) produces a 908 bend in a very rough rope tied to a wall, and its lower end rests on a rough floor (Fig. P12.51). Let ms repre-sent the coefficient of static friction between beam and floor. Assume ms is less than the cotangent of u. (a) Find an expression for the maximum mass M that can be suspended from the top before the beam slips. Determine (b) the magnitude of the reaction force at the floor and (c) the magnitude of the force exerted by the beam on the rope at P in terms of m, M, and ms.

P

m

u

M

Figure P12.51

52. The large quadriceps muscle in the upper leg terminates at its lower end in a tendon attached to the upper end of the tibia (Fig. P12.52a, page 386). The forces on the lower leg when the leg is extended are modeled as in Figure P12.52b, where T

S is the force in the tendon, F

Sg,leg is

the gravitational force acting on the lower leg, and FS

g,foot is the gravitational force acting on the foot. Find T when the tendon is at an angle of f 5 25.08 with the tibia, assuming Fg,leg 5 30.0 N, Fg,foot 5 12.5 N, and the leg is extended at an angle u  5 40.08 with respect to the vertical. Also assume the center of gravity of the

S

BIO

is pivoted with a frictionless pin at A and rests against a smooth support at B. Find the reaction forces at (a) point A and (b) point B.

B

A

2.00 m6.00 m

m2

1.00 m Sm1g

Figure P12.47

48. Assume a person bends forward to lift a load “with his back” as shown in Figure P12.48a. The spine pivots mainly at the fifth lumbar vertebra, with the princi-pal supporting force provided by the erector spinalis muscle in the back. To see the magnitude of the forces involved, consider the model shown in Figure P12.48b for a person bending forward to lift a 200-N object. The spine and upper body are represented as a uniform hor-izontal rod of weight 350 N, pivoted at the base of the spine. The erector spinalis muscle, attached at a point two-thirds of the way up the spine, maintains the posi-tion of the back. The angle between the spine and this muscle is u 5 12.08. Find (a) the tension T in the back muscle and (b) the compressional force in the spine. (c) Is this method a good way to lift a load? Explain your answer, using the results of parts (a) and (b). (d) Can you suggest a better method to lift a load?

a b

Pivot

Back muscle

Ry

Rx

12.0!

200 N350 N

TS

Figure P12.48

49. A 10 000-N shark is supported by a rope attached to a 4.00-m rod that can pivot at the base. (a) Calculate the tension in the cable between the rod and the wall, assuming the cable is holding the system in the position shown in Fig-ure P12.49. Find (b) the hori-zontal force and (c) the verti-cal force exerted on the base of the rod. Ignore the weight of the rod.

50. Why is the following situation impossible? A worker in a factory pulls a cabinet across the floor using a rope as

Q/CBIO

M 20.0!

60.0! 10 000 N

Figure P12.49

8

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Equations

xf = xi + vit+ 12at2

vf = vi + at

vavg =vi+vf

2

v2f = v2i + 2a∆x

xf = xi + vavgt

ω = 2πT

v = rω

at = rα

ac = v2

r

vf = vi + ve ln(mi

mf

)Thrust = ve

dmdt

rCM = 1Mtot

∑imiri

I =∑

imir2i

I ′ = ICM +MD2

T = 2π√

mk

T = 2π√

Lg

T = 2π√

Imgd

x = A cos(ωt+ φ)

K = 12mv2

Ug = mgy

Us = 12kx2

UG = −Gm1m2

r

W =∫τ · dθ =

∫F · ds

P = τ · ω = F · vFx = − dU

dx

R = −bvv(t) = vT (1 − e−bt/m)

R = 12DρAv2

v(t) = vT tanh(

gvTt)

dAdt

= L2Mp

T 2 =(

4π2

GM

)a3

p = mv

F = dpdt

τ = r× F

L = r× p

τ = dLdt

I =∫F(t) dt

∆L =∫τ dt

L = Iω

L = mvR

ωp = Mg rCM

Iω

Fk = µkn

Fs,max = µsn

F = −kxFG = −Gm1m2

r2r̂

E = −Gm1m2

2a

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Moments of Inertia

All objects listed here have mass M .

Thin rod, length L, axis through CM perpendicular to rod: I = 112ML2

Solid sphere, radius R, axis through CM: I = 25MR2

Cylinder or disc, radius R, axis through CM: I = 12MR2

Thin ring, radius R, axis through CM: I = MR2

Trigonometric Identities

sin2 θ + cos2 θ = 1

sin(2θ) = 2 sin(θ) cos(θ).

cos(2θ) = cos2 θ − sin2 θ

sin(α± β) = sinα cos β ± cosα sin β

cos(α± β) = cosα cos β ∓ sinα sin β

cosα cos β = 12[cos(α− β) + cos(α + β)]

sinα sin β = 12[cos(α− β) − cos(α + β)]

sinα cos β = 12[sin(α + β) + sin(α− β)]

sin(θ + π

2

)= cos θ

cos(θ + π

2

)= − sin θ

sec θ := 1cos θ

csc θ := 1sin θ

cot θ := 1tan θ

12