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Home Work 6 – Chapter 6

5. A 2.5 kg block is initially at rest on a horizontal surface. A horizontal force

of magnitude

6.0 N and a vertical force are then applied to the block (Fig. 6-17). The coefficients of friction for the block and surface are μs = 0.40 and μk = 0.25. Determine the magnitude of the frictional force acting on the block if the magnitude of 12 N.

is (a) 8.0 N, (b) 10 N, and (c)

Figure 6-17 Problem 5.

Solutions: In addition to the forces already shown in Fig. 6-17, a free-body diagram would include an upward

normal force FN exerted by the floor on the block, a downward mg representing the gravitational pull

exerted by Earth, and an assumed-leftward f for the kinetic or static friction. We choose +x rightwards and +y upwards. We apply Newton’s second law to these axes:

F f ma P FN mg 0 where F = 6.0 N and m = 2.5 kg is the mass of the block. (a) In this case, P = 8.0 N leads to FN = (2.5 kg)(9.8 m/s2) – 8.0 N = 16.5 N. Using Eq. 6-1, this implies f s ,max s FN 6.6 N , which is larger than the 6.0 N rightward force – so the block (which was initially at rest) does not move. Putting a = 0 into the first of our equations above yields a static friction force of f = P = 6.0 N. (b) In this case, P = 10 N, the normal force is FN = (2.5 kg)(9.8 m/s2) – 10 N = 14.5 N.

Using Eq. 6-1, this implies f s ,max s FN 5.8 N , which is less than the 6.0 N rightward force – so the block does move. Hence, we are dealing not with static but with kinetic friction, which Eq. 6-2 reveals to be f k k FN 3.6 N . (c) In this last case, P = 12 N leads to FN = 12.5 N and thus to f s ,max s FN 5.0 N , which (as expected) is less than the 6.0 N rightward force – so the block moves. The kinetic friction force, then, is

f k k FN 3.1 N .

10. Figure 6-20 shows an initially stationary block of mass m on a floor. A force of magnitude 0.500mg is then applied at upward angle θ = 20°. What is the magnitude of the acceleration of the block across the floor if the friction coefficients are (a) μs = 0.600 and μk = 0.500 and (b) μs = 0.400 and μk = 0.300?

Figure 6-20 Problem 10.

Solutions: (a) The free-body diagram for the block is shown below, with F being the force applied to the block, FN

the normal force of the floor on the block, mg the force of gravity, and f the force of friction. We take the +x direction to be horizontal to the right and the +y direction to be up. The equations for the x and the y components of the force according to Newton’s second law are:

Fx F cos f ma Fy F sin FN mg 0

Now f =kFN, and the second equation gives FN = mg – Fsin, which yields f k (mg F sin ) . This expression is substituted for f in the first equation to obtain

F cos – k (mg – F sin ) = ma, so the acceleration is

a

F cos k sin k g . m

(a) If s 0.600 and k 0.500, then the magnitude of f has a maximum value of

f s ,max s FN (0.600)(mg 0.500mg sin 20) 0.497mg. On the other hand, F cos 0.500mg cos 20 0.470mg. Therefore, F cos f s ,max and the block remains stationary with a 0 . (b) If s 0.400 and k 0.300, then the magnitude of f has a maximum value of

f s ,max s FN (0.400)(mg 0.500mg sin 20) 0.332mg. In this case, F cos 0.500mg cos 20 0.470mg f s ,max . Therefore, the acceleration of the block is

F cos k sin k g m (0.500)(9.80 m/s 2 ) cos 20 (0.300)sin 20 (0.300)(9.80 m/s 2 ) 2.17 m/s 2 .

a

16. A loaded penguin sled weighing 80 N rests on a plane inclined at angle θ = 20° to the horizontal (Fig. 6-23). Between the sled and the plane, the coefficient of static friction is 0.25, and the coefficient of kinetic friction is 0.15. (a) What is the least magnitude of the force parallel to the plane, that will prevent the sled from slipping down the plane? (b) What is the minimum magnitude F that will start the sled moving up the plane? (c) What value of F is required to move the sled up the plane at constant velocity?

Figure 6-23 Problems 16 and 22.

Solutions:

(a) In this situation, we take f s to point uphill and to be equal to its maximum value, in which case fs, max = s FN applies, where s = 0.25. Applying Newton’s second law to the block of mass m = W/g = 8.2 kg, in the x and y directions, produces

Fmin 1 mg sin f s , max ma 0 FN mg cos 0 which (with = 20°) leads to

Fmin 1 mg sin s cos 8.6 N.

(b) Now we take f s to point downhill and to be equal to its maximum value, in which case fs, max = sFN applies, where s = 0.25. Applying Newton’s second law to the block of mass m = W/g = 8.2 kg, in the x and y directions, produces

Fmin 2 mg sin f s , max ma 0 FN mg cos 0 which (with = 20°) leads to

Fmin 2 mg sin s cos 46 N. A value slightly larger than the “exact” result of this calculation is required to make it accelerate uphill, but since we quote our results here to two significant figures, 46 N is a “good enough” answer. (c) Finally, we are dealing with kinetic friction (pointing downhill), so that

0 F mg sin f k ma 0 FN mg cos along with fk = kFN (where k = 0.15) brings us to

(

)

29. In Fig. 6-34, blocks A and B have weights of 44 N and 22 N, respectively. (a) Determine the minimum weight of block C to keep A from sliding if μs between A and the table is 0.20. (b) Block Csuddenly is lifted off A. What is the acceleration of block A if μk between A and the table is 0.15?

Figure 6-34 Problem 29.

Solutions: (a) Free-body diagrams for the blocks A and C, considered as a single object, and for the block B are shown below.

T is the magnitude of the tension force of the rope, FN is the magnitude of the normal force of the table on block A, f is the magnitude of the force of friction, WAC is the combined weight of blocks A and C (the

magnitude of force Fg AC shown in the figure), and WB is the weight of block B (the magnitude of force

Fg B shown). Assume the blocks are not moving. For the blocks on the table we take the x axis to be to

the right and the y axis to be upward. From Newton’s second law, we have

x component:

T–f=0

y component:

FN – WAC = 0.

For block B take the downward direction to be positive. Then Newton’s second law for that block is WB – T = 0. The third equation gives T = WB and the first gives f = T = WB. The second equation gives FN = WAC. If sliding is not to occur, f must be less than s FN, or WB < s WAC. The smallest that WAC can be with the blocks still at rest is WAC = WB/s = (22 N)/(0.20) = 110 N. Since the weight of block A is 44 N, the least weight for C is (110 – 44) N = 66 N. (b) The second law equations become T – f = (WA/g)a FN – WA = 0 WB – T = (WB/g)a. In addition, f = kFN. The second equation gives FN = WA, so f = kWA. The third gives T = WB – (WB/g)a. Substituting these two expressions into the first equation, we obtain WB – (WB/g)a – kWA = (WA/g)a. Therefore, 2 g WB kWA (9.8 m/s ) 22 N 0.15 44 N a 2.3 m/s 2 . WA WB 44 N + 22 N

49. In Fig. 6-39, a car is driven at constant speed over a circular hill and then into a circular valley with the same radius. At the top of the hill, the normal force on the driver from the car seat is 0. The driver's mass is 70.0 kg. What is the magnitude of the normal force on the driver from the seat when the car passes through the bottom of the valley?

Figure 6-39 Problem 49.

Solutions: At the top of the hill, the situation is similar to that of Sample Problem – “Vertical circular loop, Diavolo,” but with the normal force direction reversed. Adapting Eq. 6-19, we find FN = m(g – v2/R). Since FN = 0 there (as stated in the problem) then v2 = gR. Later, at the bottom of the valley, we reverse both the normal force direction and the acceleration direction (from what is shown in the Sample Problem) and adapt Eq. 6-19 accordingly. Thus we obtain FN = m(g + v2/R) = 2mg = 1372 N 1.37 103 N.

57. A puck of mass m = 1.50 kg slides in a circle of radius r = 20.0 cm on a frictionless table while attached to a hanging cylinder of mass M = 2.50 kg by means of a cord that extends through a hole in the table (Fig. 6-43). What speed keeps the cylinder at rest?

Figure 6-43 Problem 57.

Solutions: For the puck to remain at rest the magnitude of the tension force T of the cord must equal the gravitational force Mg on the cylinder. The tension force supplies the centripetal force that keeps the puck in its circular orbit, so T = mv2/r. Thus Mg = mv2/r. We solve for the speed:

v

Mgr (2.50 kg)(9.80 m/s 2 )(0.200 m) 1.81 m/s. m 1.50 kg

70. Figure 6-53 shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed. (The cord sweeps out a cone as the bob rotates.) The bob has a mass of 0.040 kg, the string has length L = 0.90 m and negligible mass, and the bob follows a circular path of circumference 0.94 m. What are (a) the tension in the string and (b) the period of the motion?

Figure 6-53 Problem 70.

Solutions: (a) We note that R (the horizontal distance from the bob to the axis of rotation) is the circumference of the circular path divided by 2; therefore, R = 0.94/2= 0.15 m. The angle that the cord makes with the horizontal is now easily found:

= cos1(R/L) = cos1(0.15 m/0.90 m) = 80º.

The vertical component of the force of tension in the string is Tsin and must equal the downward pull of gravity (mg). Thus,

T

mg 0.40 N . sin

Note that we are using T for tension (not for the period). (b) The horizontal component of that tension must supply the centripetal force (Eq. 6-18), so we have Tcos = mv2/R. This gives speed v = 0.49 m/s. This divided into the circumference gives the time for one revolution: 0.94/0.49 = 1.9 s.

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5. A 2.5 kg block is initially at rest on a horizontal surface. A horizontal force

of magnitude

6.0 N and a vertical force are then applied to the block (Fig. 6-17). The coefficients of friction for the block and surface are μs = 0.40 and μk = 0.25. Determine the magnitude of the frictional force acting on the block if the magnitude of 12 N.

is (a) 8.0 N, (b) 10 N, and (c)

Figure 6-17 Problem 5.

Solutions: In addition to the forces already shown in Fig. 6-17, a free-body diagram would include an upward

normal force FN exerted by the floor on the block, a downward mg representing the gravitational pull

exerted by Earth, and an assumed-leftward f for the kinetic or static friction. We choose +x rightwards and +y upwards. We apply Newton’s second law to these axes:

F f ma P FN mg 0 where F = 6.0 N and m = 2.5 kg is the mass of the block. (a) In this case, P = 8.0 N leads to FN = (2.5 kg)(9.8 m/s2) – 8.0 N = 16.5 N. Using Eq. 6-1, this implies f s ,max s FN 6.6 N , which is larger than the 6.0 N rightward force – so the block (which was initially at rest) does not move. Putting a = 0 into the first of our equations above yields a static friction force of f = P = 6.0 N. (b) In this case, P = 10 N, the normal force is FN = (2.5 kg)(9.8 m/s2) – 10 N = 14.5 N.

Using Eq. 6-1, this implies f s ,max s FN 5.8 N , which is less than the 6.0 N rightward force – so the block does move. Hence, we are dealing not with static but with kinetic friction, which Eq. 6-2 reveals to be f k k FN 3.6 N . (c) In this last case, P = 12 N leads to FN = 12.5 N and thus to f s ,max s FN 5.0 N , which (as expected) is less than the 6.0 N rightward force – so the block moves. The kinetic friction force, then, is

f k k FN 3.1 N .

10. Figure 6-20 shows an initially stationary block of mass m on a floor. A force of magnitude 0.500mg is then applied at upward angle θ = 20°. What is the magnitude of the acceleration of the block across the floor if the friction coefficients are (a) μs = 0.600 and μk = 0.500 and (b) μs = 0.400 and μk = 0.300?

Figure 6-20 Problem 10.

Solutions: (a) The free-body diagram for the block is shown below, with F being the force applied to the block, FN

the normal force of the floor on the block, mg the force of gravity, and f the force of friction. We take the +x direction to be horizontal to the right and the +y direction to be up. The equations for the x and the y components of the force according to Newton’s second law are:

Fx F cos f ma Fy F sin FN mg 0

Now f =kFN, and the second equation gives FN = mg – Fsin, which yields f k (mg F sin ) . This expression is substituted for f in the first equation to obtain

F cos – k (mg – F sin ) = ma, so the acceleration is

a

F cos k sin k g . m

(a) If s 0.600 and k 0.500, then the magnitude of f has a maximum value of

f s ,max s FN (0.600)(mg 0.500mg sin 20) 0.497mg. On the other hand, F cos 0.500mg cos 20 0.470mg. Therefore, F cos f s ,max and the block remains stationary with a 0 . (b) If s 0.400 and k 0.300, then the magnitude of f has a maximum value of

f s ,max s FN (0.400)(mg 0.500mg sin 20) 0.332mg. In this case, F cos 0.500mg cos 20 0.470mg f s ,max . Therefore, the acceleration of the block is

F cos k sin k g m (0.500)(9.80 m/s 2 ) cos 20 (0.300)sin 20 (0.300)(9.80 m/s 2 ) 2.17 m/s 2 .

a

16. A loaded penguin sled weighing 80 N rests on a plane inclined at angle θ = 20° to the horizontal (Fig. 6-23). Between the sled and the plane, the coefficient of static friction is 0.25, and the coefficient of kinetic friction is 0.15. (a) What is the least magnitude of the force parallel to the plane, that will prevent the sled from slipping down the plane? (b) What is the minimum magnitude F that will start the sled moving up the plane? (c) What value of F is required to move the sled up the plane at constant velocity?

Figure 6-23 Problems 16 and 22.

Solutions:

(a) In this situation, we take f s to point uphill and to be equal to its maximum value, in which case fs, max = s FN applies, where s = 0.25. Applying Newton’s second law to the block of mass m = W/g = 8.2 kg, in the x and y directions, produces

Fmin 1 mg sin f s , max ma 0 FN mg cos 0 which (with = 20°) leads to

Fmin 1 mg sin s cos 8.6 N.

(b) Now we take f s to point downhill and to be equal to its maximum value, in which case fs, max = sFN applies, where s = 0.25. Applying Newton’s second law to the block of mass m = W/g = 8.2 kg, in the x and y directions, produces

Fmin 2 mg sin f s , max ma 0 FN mg cos 0 which (with = 20°) leads to

Fmin 2 mg sin s cos 46 N. A value slightly larger than the “exact” result of this calculation is required to make it accelerate uphill, but since we quote our results here to two significant figures, 46 N is a “good enough” answer. (c) Finally, we are dealing with kinetic friction (pointing downhill), so that

0 F mg sin f k ma 0 FN mg cos along with fk = kFN (where k = 0.15) brings us to

(

)

29. In Fig. 6-34, blocks A and B have weights of 44 N and 22 N, respectively. (a) Determine the minimum weight of block C to keep A from sliding if μs between A and the table is 0.20. (b) Block Csuddenly is lifted off A. What is the acceleration of block A if μk between A and the table is 0.15?

Figure 6-34 Problem 29.

Solutions: (a) Free-body diagrams for the blocks A and C, considered as a single object, and for the block B are shown below.

T is the magnitude of the tension force of the rope, FN is the magnitude of the normal force of the table on block A, f is the magnitude of the force of friction, WAC is the combined weight of blocks A and C (the

magnitude of force Fg AC shown in the figure), and WB is the weight of block B (the magnitude of force

Fg B shown). Assume the blocks are not moving. For the blocks on the table we take the x axis to be to

the right and the y axis to be upward. From Newton’s second law, we have

x component:

T–f=0

y component:

FN – WAC = 0.

For block B take the downward direction to be positive. Then Newton’s second law for that block is WB – T = 0. The third equation gives T = WB and the first gives f = T = WB. The second equation gives FN = WAC. If sliding is not to occur, f must be less than s FN, or WB < s WAC. The smallest that WAC can be with the blocks still at rest is WAC = WB/s = (22 N)/(0.20) = 110 N. Since the weight of block A is 44 N, the least weight for C is (110 – 44) N = 66 N. (b) The second law equations become T – f = (WA/g)a FN – WA = 0 WB – T = (WB/g)a. In addition, f = kFN. The second equation gives FN = WA, so f = kWA. The third gives T = WB – (WB/g)a. Substituting these two expressions into the first equation, we obtain WB – (WB/g)a – kWA = (WA/g)a. Therefore, 2 g WB kWA (9.8 m/s ) 22 N 0.15 44 N a 2.3 m/s 2 . WA WB 44 N + 22 N

49. In Fig. 6-39, a car is driven at constant speed over a circular hill and then into a circular valley with the same radius. At the top of the hill, the normal force on the driver from the car seat is 0. The driver's mass is 70.0 kg. What is the magnitude of the normal force on the driver from the seat when the car passes through the bottom of the valley?

Figure 6-39 Problem 49.

Solutions: At the top of the hill, the situation is similar to that of Sample Problem – “Vertical circular loop, Diavolo,” but with the normal force direction reversed. Adapting Eq. 6-19, we find FN = m(g – v2/R). Since FN = 0 there (as stated in the problem) then v2 = gR. Later, at the bottom of the valley, we reverse both the normal force direction and the acceleration direction (from what is shown in the Sample Problem) and adapt Eq. 6-19 accordingly. Thus we obtain FN = m(g + v2/R) = 2mg = 1372 N 1.37 103 N.

57. A puck of mass m = 1.50 kg slides in a circle of radius r = 20.0 cm on a frictionless table while attached to a hanging cylinder of mass M = 2.50 kg by means of a cord that extends through a hole in the table (Fig. 6-43). What speed keeps the cylinder at rest?

Figure 6-43 Problem 57.

Solutions: For the puck to remain at rest the magnitude of the tension force T of the cord must equal the gravitational force Mg on the cylinder. The tension force supplies the centripetal force that keeps the puck in its circular orbit, so T = mv2/r. Thus Mg = mv2/r. We solve for the speed:

v

Mgr (2.50 kg)(9.80 m/s 2 )(0.200 m) 1.81 m/s. m 1.50 kg

70. Figure 6-53 shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed. (The cord sweeps out a cone as the bob rotates.) The bob has a mass of 0.040 kg, the string has length L = 0.90 m and negligible mass, and the bob follows a circular path of circumference 0.94 m. What are (a) the tension in the string and (b) the period of the motion?

Figure 6-53 Problem 70.

Solutions: (a) We note that R (the horizontal distance from the bob to the axis of rotation) is the circumference of the circular path divided by 2; therefore, R = 0.94/2= 0.15 m. The angle that the cord makes with the horizontal is now easily found:

= cos1(R/L) = cos1(0.15 m/0.90 m) = 80º.

The vertical component of the force of tension in the string is Tsin and must equal the downward pull of gravity (mg). Thus,

T

mg 0.40 N . sin

Note that we are using T for tension (not for the period). (b) The horizontal component of that tension must supply the centripetal force (Eq. 6-18), so we have Tcos = mv2/R. This gives speed v = 0.49 m/s. This divided into the circumference gives the time for one revolution: 0.94/0.49 = 1.9 s.

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