Mechanics

An airplane flies in a loop (a circular path in a vertical plane) of radius 150 m. The pilot’s head always points toward the center of the loop. The speed of the airplane is not constant; the airplane goes slowest at the top of the loop and fastest at the bottom. 

(a) What is the speed of the airplane at the top of the loop, where the pilot feels weightless? 

(b) What is the apparent weight of the pilot at the bottom of the loop, where the speed of the airplane is $280\text{\hspace{0.17em}km/h}$? His true weight is 700 N.

You tie a cord to a pail of water and swing the pail in a vertical circle of radius $0.600\text{\hspace{0.17em}m}$. What minimum speed must you give the pail at the highest point of the circle to avoid spilling water?

A bowling ball weighing 71.2 N (16.0 lb) is attached to the ceiling by a 3.80-m rope. The ball is pulled to one side and released; it then swings back and forth as a pendulum. As the rope swings through the vertical, the speed of the bowling ball is $4.20\text{\hspace{0.17em}m/s}$. At this instant, what are 

(a) the acceleration of the bowling ball, in magnitude and direction, and 

(b) the tension in the rope?

A $6.00\text{\hspace{0.17em}kg}$ box sits on a ramp that is inclined at $37{.0}^{\circ }$ above the horizontal. The coefficient of kinetic friction between the box and the ramp is ${\mu }_k=0.30$. What horizontal force is required to move the box up the incline with a constant acceleration of  $3.60{\text{\hspace{0.17em}m/s}}^{\text{2}}$?

The “Giant Swing” at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. Each arm supports a seat suspended from a cable 5.00m long, and the upper end of the cable is fastened to the arm at a point 3.00m from the central shaft (Fig. E5.50). (a) Find the time of one revolution of the swing if the cable supporting a seat makes an angle of $\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{°}$ with the vertical. (b) Does the angle depend on the weight of the passenger for a given rate of revolution?

Question: In another version of the “Giant Swing” (see Exercise 5.50), the seat is connected to two cables, one of which is horizontal (Fig. E5.51). The seat swings in a horizontal circle at a rate of 28.0rpm. If the seat weighs 255N and an 825N person is sitting in it, find the tension in each cable.

A small button placed on a horizontal rotating platform with diameter 0.520 m will revolve with the platform when it is brought up to a speed of 40.0rev/min , provided the button is no more than 0.220 m from the axis. (a) What is the coefficient of static friction between the button and the platform? (b) How far from the axis can the button be placed, without slipping, if the platform rotates at  60.0 rev/min ?

Question: A small button placed on a horizontal rotating platform with diameter 0.520 m will revolve with the platform when it is brought up to a speed of $\mathbf{40}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{rev/min}}$, provided the button is no more than 0.220 m from the axis. (a) What is the coefficient of static friction between the button and the platform? (b) How far from the axis can the button be placed, without slipping, if the platform rotates at $\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{rev/min}}$?

One problem for humans living in outer space is that they are apparently weightless. One way around this problem is to design a space station that spins about its center at a constant rate. This creates “artificial gravity” at the outside rim of the station. (a) If the diameter of the space station is , how many revolutions per minute are needed for the “artificial gravity” acceleration to be $\mathbf{9}\mathbf{.}\mathbf{80}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$ ? (b) If the space station is a waiting area for travelers going to Mars, it might be desirable to simulate the acceleration due to gravity on the Martian surface (${\mathbf{\text{3.70 m/s}}}^{\mathbf{2}}$). How many revolutions per minute are needed in this case?

The Cosmo Clock 21 Ferris wheel in Yokohama, Japan, has a diameter of 100 m. Its name comes from its 60 arms, each of which can function as a second hand (so that it makes one revolution every 60.0 s). 

(a) Find the speed of the passengers when the Ferris wheel is rotating at this rate. (b) A passenger weighs 882 Nat the weight-guessing booth on the ground. What is his apparent weight at the highest and at the lowest point on the Ferris wheel? (c) What would be the time for one revolution if the passenger’s apparent weight at the highest point were zero? 

(d) What then would be the passenger’s apparent weight at the lowest point?

A 50.0-kg stunt pilot who has been diving her airplane vertically pulls out of the dive by changing her course to a circle in a vertical plane. (a) If the plane’s speed at the lowest point of the circle is 95.0m/s , what is the minimum radius of the circle so that the acceleration at this point will not exceed  4.00 g? (b) What is the apparent weight of the pilot at the lowest point of the pullout?

An adventurous archaeologist crosses between two rock cliffs by slowly going hand over hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope (Fig. P5.60). The rope will break if the tension in it exceeds $\mathbf{2}\mathbf{.}\mathbf{50}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathit{N}$ , and our hero’s mass is 90.0kg . (a) If the angle $\mathit{\theta }$ is $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{°}$ , what is the tension in the rope? (b) What is the smallest value  can have if the rope is not to break?

Question: A $\mathbf{3}.\mathbf{00}-\mathbf{kg}$ box that is several hundred meters above the earth’s surface is suspended from the end of a short vertical rope of negligible mass. A time-dependent upward force is applied to the upper end of the rope and results in a tension in the rope of $\mathbf{T}\left(\mathbf{t}\right)=\left(\mathbf{36}.\mathbf{0} \mathbf{N}/\mathbf{s}\right)\mathbf{t}$. The box is at rest at t = 0. The only forces on the box are the tension in the rope and gravity. (a) What is the velocity of the box at (i) $\mathbf{t}=\mathbf{1}.\mathbf{00} \mathbf{s}$ and (ii) $\mathbf{t}=\mathbf{3}.\mathbf{00} \mathbf{s}$? (b) What is the maximum distance that the box descends below its initial position? (c) At what value of   does the box return to its initial position?

Two ropes are connected to a steel cable that supports a hanging weight (Fig. P5.61). (a) Draw a free-body diagram showing all of the forces acting at the knot that connects the two ropes to the steel cable. Based on your diagram, which of the two ropes will have the greater tension? (b) If the maximum tension either rope can sustain without breaking is 5000N , determine the maximum value of the hanging weight that these ropes can safely support. Ignore the weight of the ropes and of the steel cable.

a worker lifts a weight w  by pulling down on a rope with a force $\overrightarrow{\mathbf{F}}$ . The upper pulley is attached to the ceiling by a chain, and the lower pulley is attached to the weight by another chain. Draw one or more free-body diagrams to find the tension in each chain and the magnitude of $\overrightarrow{\mathbf{F}}$ , in terms of w , if the weight is lifted at constant speed. Assume that the rope, pulleys, and chains have negligible weights.

In a repair shop a truck engine that has mass $\mathbf{409}\mathbf{ }\mathbf{\text{kg}}$ is held in place by four light cables (Fig. P5.63). Cable A is horizontal, cables B and D are vertical, and cable C makes an angle of $\mathbf{37}\mathbf{.}\mathbf{1}\mathbf{°}$ with a vertical wall. If the tension in cable A is $\mathbf{722}\mathbf{ }\mathbf{\text{N}}$, what are the tensions in cables B and C?

In a repair shop a truck engine that has mass 409 kg is held in place by four light cables (Fig. P5.63). Cable A is horizontal, cables B and D are vertical, and cable C makes an angle of $\mathbf{37}\mathbf{.}\mathbf{1}\mathbf{°}$ with a vertical wall. If the tension in cable A is 722 N, what are the tensions in cables B and C?

A horizontal wire holds a solid uniform ball of mass in place on a tilted ramp that rises $\mathbf{35}\mathbf{.}\mathbf{0}\mathbf{°}$ above the horizontal. The surface of this ramp is perfectly smooth, and the wire is directed away from the center of the ball (Fig. P5.64). 

(a) Draw a freebody diagram of the ball. 

(b) How hard does the surface of the ramp push on the ball? 

(c) What is the tension in the wire?

A solid uniform 45.0-kg ball of diameter 32.0 cm is supported against a vertical, frictionless wall by a thin 30.0-cm wire of negligible mass (Fig. P5.65). 

(a) Draw a free-body diagram for the ball, and use the diagram to find the tension in the wire. 

(b) How hard does the ball push against the wall?

A box is sliding with a constant speed of 4.00m/s in the +x -direction on a horizontal, frictionless surface. At x = 0 the box encounters a rough patch of the surface, and then the surface becomes even rougher. Between x = 0 and x = 2.00 m, the coefficient of kinetic friction between the box and the surface is 0.200; between x = 2.00 m and x = 4.00 m , it is 0.400 . (a) What is the  -coordinate of the point where the box comes to rest? (b) How much time does it take the box to come to rest after it first encounters the rough patch at x = 0 ?

When you do a chin-up, you raise your chin just over a bar (the chinning bar), supporting yourself with only your arms. Typically, the body below the arms is raised by about 30 cm in a time of 1.0 s , starting from rest. Assume that the entire body of a -680-N  person doing chin-ups is raised by 30 cm, and that half the 1.0 s  is spent accelerating upward and the other half accelerating downward, uniformly in both cases. Draw a free-body diagram of the person’s body, and use it to find the force his arms must exert on him during the accelerating part of the chin-up.

A 2.00 - kg box is suspended from the end of a light vertical rope. A time-dependent force is applied to the upper end of the rope, and the box moves upward with a velocity magnitude that varies in time according to $\mathbf{v}\left(t\right)\mathbf{=}\left(2.00 m/s^2\right)\mathbf{t}\mathbf{+}\left(0.600 m/s^3\right){\mathbf{t}}^{\mathbf{2}}$. What is the tension in the rope when the velocity of the box is ?

A 3.00-kg  box that is several hundred meters above the earth’s surface is suspended from the end of a short vertical rope of negligible mass. A time-dependent upward force is applied to the upper end of the rope and results in a tension in the rope of $\mathbf{T}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{36}\mathbf{.}\mathbf{0}\mathbf{\hspace{0.33em}}\mathbf{N}\mathbf{/}\mathbf{s}\mathbf{)}\mathbf{t}$ . The box is at rest at t=0 . The only forces on the box are the tension in the rope and gravity. (a) What is the velocity of the box at (i) t = 1.00 s and (ii) t = 3.00 s ? (b) What is the maximum distance that the box descends below its initial position? (c) At what value of t does the box return to its initial position?

A 5.00 - kg box sits at rest at the bottom of a ramp that is 8.00 m long and is inclined at $\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{°}$ above the horizontal. The coefficient of kinetic friction is ${\mathit{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{40}$, and the coefficient of static friction is ${\mathit{\mu }}_{\mathbf{s}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{43}$. What constant force F , applied parallel to the surface of the ramp, is required to push the box to the top of the ramp in a time of 6.00 s ?

Question: Two boxes connected by a light horizontal rope are on a horizontal surface (Fig. E5.37). The coefficient of kinetic friction between each box and the surface is ${\mathit{\mu }}_{\mathbf{K}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{30}$. Box has mass 5.00kg, and box A has mass m. A force F with magnitude 40.0N and direction $\mathbf{53}\mathbf{.}\mathbf{1}\mathbf{°}$ above the horizontal is applied to the 5.00kg box, and both boxes move to the right with $\mathbf{a}=\mathbf{1}.\mathbf{50} \mathbf{m}/{\mathbf{s}}^{\mathbf{2}}$. (a) What is the tension T in the rope that connects the boxes? (b) What is m?

In Fig. P5.74, and $m_1=20.0\text{\hspace{0.17em}kg}$. The coefficient of kinetic friction between the block of mass $m_1$and the incline is . What must be the mass $m_2$ of the hanging block if it is to descend $12.0\text{\hspace{0.17em}m}$in the first $3.00\text{\hspace{0.17em}s}$after the system is released from rest?

An 8.00 - kg box sits on a ramp that is inclined at $\mathbf{33}\mathbf{.}\mathbf{0}\mathbf{°}$ above the horizontal. The coefficient of kinetic friction between the box and the surface of the ramp is${\mathbf{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{300}$. A constant horizontal force F = 26.0 N is applied to the box (Fig. P5.73), and the box moves down the ramp. If the box is initially at rest, what is its speed 2.00 s after the force is applied?

Question: You place a book of mass $\mathbf{5}.\mathbf{00}\mathbf{kg}$  against a vertical wall. You apply a constant force  $\overrightarrow{\mathbf{F}}$ to the book, where $\mathbf{F}\mathbf{=}\mathbf{96}\mathbf{.}\mathbf{0}\mathbf{N}$  and the force is at an angle of  $\mathbf{60}\mathbf{.}{\mathbf{0}}^{\mathbf{\circ }}$ above the horizontal (Fig. P5.75). The coefficient of kinetic friction between the book and the wall is 0.300 m . If the book is initially at rest, what is its speed after it has travelled  0.400 m up the wall?

Block A in Fig. P5.76 weighs 60.0 N. The coefficient of static friction between the block and the surface on which it rests is 0.25. The weight 12.0 N , and the system is in equilibrium. (a) Find the friction force exerted on block A. (b) Find the maximum weight w for which the system will remain in equilibrium.

A block with mass${\mathbf{m}}_{\mathbf{1}}$is placed on an inclined plane with slope angle and is connected to a hanging block with mass ${\mathbf{m}}_{\mathbf{2}}$by a cord passing over a small, frictionless pulley (Fig. P5.74). The coefficient of static friction is , and the coefficient of kinetic friction is${\mathbf{\mu }}_{\mathbf{k}}$ (a) Find the value of for which the block of mass ${\mathbf{m}}_{\mathbf{1}}$moves up the plane at constant speed once it is set in motion. (b) Find the value of for which the block of mass ${\mathbf{m}}_{\mathbf{2}}$moves down the plane at constant speed once it is set in motion. (c) For what range of values of ${\mathbf{m}}_{\mathbf{2}}$will the blocks remain at rest if they are released from rest?

The Flying Leap of a Flea. High-speed motion pictures (3500 frames/second) of a jumping 210 - ug flea yielded the data to plot the flea’s acceleration as a function of time, as shown in Fig. P5.78. (See “The Flying Leap of the Flea,” by M. Rothschild et al., Scientific American, November 1973.) This flea was about  long and jumped at a nearly vertical takeoff angle. Using the graph, (a) find the initial net external force on the flea. How does it compare to the flea’s weight? (b) Find the maximum net external force on this jumping flea. When does this maximum force occur? (c) Use the graph to find the flea’s maximum speed.

Block A in Fig. P5.79 weighs 1.20 N , and block B weighs 3.60 N . The coefficient of kinetic friction between all surfaces is 0.300. Find the magnitude of the horizontal force $\overrightarrow{\mathbf{f}}$ necessary to drag block B to the left at constant speed (a) if A rests on B and moves with it (Fig. P5.79a), (b) if A is held at rest (Fig. P5.79b). 

You are designing an elevator for a hospital. The force exerted on a passenger by the floor of the elevator is not to exceed 1.60 times the passenger’s weight. The elevator accelerates upward with constant acceleration for a distance of 3.0 m and then starts to slow down. What is the maximum speed of the elevator?

You are standing on a bathroom scale in an elevator in a tall building. Your mass is 64 kg . The elevator starts from rest and travels upward with a speed that varies with time according to $\mathbf{v}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\left(3.0 m/s^2\right)\mathbf{t}\mathbf{+}\left(0.20 m/s^3\right){\mathbf{t}}^{\mathbf{2}}$ . When t = 4.0 s , what is the reading on the bathroom scale?

Question: Traffic Court. You are called as an expert witness in a trial for a traffic violation. The facts are these: A driver slammed on his brakes and came to a stop with constant acceleration. Measurements of his tires and the skid marks on the pavement indicate that he locked his car’s wheels, the car traveled   before stopping, and the coefficient of kinetic friction between the road and his tires was 0.750 . He was charged with speeding in a 45- mi/h   zone but pleads innocent. What is your conclusion: guilty or innocent? How fast was he going when he hit his brakes?

Block A in Fig. P5.87 weighs $1.90\text{\hspace{0.17em}N}$, and block B weighs .$4.20\text{\hspace{0.17em}N}$ The coefficient of kinetic friction between all surfaces is $0.30$. Find the magnitude of the horizontal force $\overrightarrow{F}$ necessary to drag block B to the left at constant speed if A and B are connected by a light, flexible cord passing around a fixed, frictionless pulley.

A hammer is hanging by a light rope from the ceiling of a bus. The ceiling is parallel to the roadway. The bus is traveling in a straight line on a horizontal street. You observe that the hammer hangs at rest with respect to the bus when the angle between the rope and the ceiling of the bus is $\mathbf{56}\mathbf{.}\mathbf{0}\mathbf{°}$. What is the acceleration of the bus?

Question: A $\mathbf{40}.\mathbf{0}-\mathbf{kg}$ packing case is initially at rest on the floor of a $\mathbf{1500}-\mathbf{kg}$ pickup truck. The coefficient of static friction between the case and the truck floor is 0.30, and the coefficient of kinetic friction is 0.20. Before each acceleration given below, the truck is traveling due north at constant speed. Find the magnitude and direction of the friction force acting on the case (a) when the truck accelerates at $\mathbf{2}.\mathbf{20} \mathbf{m}/{\mathbf{s}}^{\mathbf{2}}$ northward and (b) when it accelerates at $\mathbf{3}.\mathbf{40} \mathbf{m}/{\mathbf{s}}^{\mathbf{2}}$ southward.

Question: If the coefficient of static friction between a table and a uniform, massive rope is ${\mathbf{\mu }}_{\mathbf{s}}$ , what fraction of the rope can hang over the edge of the table without the rope sliding?

Two identical  15.0 - kg balls, each  25.0 cm in diameter, are suspended by two  35.0 - cm wires (Fig. P5.85). The entire apparatus is supported by a single  18.0 - cm wire, and the surfaces of the balls are perfectly smooth. (a) Find the tension in each of the three wires. (b) How hard does each ball push on the other one?

Question: Two identical   balls, each   in diameter, are suspended by two   wires (Fig. P5.85). The entire apparatus is supported by a single   wire, and the surfaces of the balls are perfectly smooth. (a) Find the tension in each of the three wires. (b) How hard does each ball push on the other one?

A  12.0 kg box rests on the level bed of a truck. The coefficients of friction between the box and bed are ${\mathbf{\mu }}_{\mathbf{s}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{19}\mathbf{\text{ and }}{\mathbf{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{15}$ . The truck stops at a stop sign and then starts to move with an acceleration of 2.20 $\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$ . If the box is   from the rear of the truck when the truck starts, how much time elapses before the box falls off the truck? How far does the truck travel in this time?

Question: A 12.0kg box rests on the level bed of a truck. The coefficients of friction between the box and bed are ${\mathit{\mu }}_{\mathbf{s}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{19}$ and ${\mathit{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{15}$. The truck stops at a stop sign and then starts to move with an acceleration of $\mathbf{2}.\mathbf{20} \mathbf{m}/{\mathbf{s}}^{\mathbf{2}}$. If the box is 1.80m from the rear of the truck when the truck starts, how much time elapses before the box falls off the truck? How far does the truck travel in this time?

Block A in Fig. P5.89 has mass 4.00 kg, and block B has mass 12.00 kg. The coefficient of kinetic friction between block B and the horizontal surface is 0.25. (a) What is the mass of block C if block B is moving to the right and speeding up with an acceleration of $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$? (b) What is the tension in each cord when block B has this acceleration?

Question: Block A in Fig. P5.89 has mass , and block B has mass . The coefficient of kinetic friction between block B and the horizontal surface is . (a) What is the mass of block C if block B is moving to the right and speeding up with an acceleration of ? (b) What is the tension in each cord when block B has this acceleration?

Two blocks connected by a cord passing over a small, frictionless pulley rest on frictionless planes (Fig. P5.90). (a) Which way will the system move when the blocks are released from rest? (b) What is the acceleration of the blocks? (c) What is the tension in the cord?

Question: Two blocks connected by a cord passing over a small, frictionless pulley rest on frictionless planes (Fig. P5.90). (a) Which way will the system move when the blocks are released from rest? (b) What is the acceleration of the blocks? (c) What is the tension in the cord?

In terms of m₁, m₂, and g, find the acceleration of each block in Fig. P5.91. There is no friction anywhere in the system.

Block B, with mass 5.00 kg, rests on block A, with mass 8.00 kg, which in turn is on a horizontal tabletop (Fig. P5.92). There is no friction between block A and the tabletop, but the coefficient of static friction between blocks A and B is 0.750. A light string attached to block A passes over a frictionless, massless pulley, and block C is suspended from the other end of the string. What is the largest mass that block C can have so that blocks A and B still slide together when the system is released from rest?

A curve with a 120-m radius on a level road is banked at the correct angle for a speed of 20 m/s. If an automobile rounds this curve at 30 m/s, what is the minimum coefficient of static friction needed between tires and road to prevent skidding?