Chapter 4

A warehouse worker pushes a crate along the floor, as shown in Figure \(4.33,\) by a force of 10 \(\mathrm{N}\) that points downward at an angle of \(45^{\circ}\) below the horizontal. Find the horizontal and vertical components of the push.

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is \(60.0^{\circ} .\) If \(\operatorname{dog} A\) exerts a force of 270 \(\mathrm{N}\) and dog \(B\) exerts a force of \(300 \mathrm{N},\) find the magnitude of the resultant force and the angle it makes with dog \(A\) 's rope.

A box rests on a frozen pond, which serves as a frictionless horizontal surface. If a fisherman applies a horizontal force with magnitude 48.0 \(\mathrm{N}\) to the box and produces an acceleration of magnitude \(3.00 \mathrm{m} / \mathrm{s}^{2},\) what is the mass of the box?

In outer space, a constant net force of magnitude 140 \(\mathrm{N}\) is exerted on a 32.5 \(\mathrm{kg}\) probe initially at rest. (a) What acceleration does this force produce? (b) How far does the probe travel in 10.0 \(\mathrm{s} ?\)

A \(\mathrm{A} 68.5 \mathrm{kg}\) skater moving initially at 2.40 \(\mathrm{m} / \mathrm{s}\) on rough horizontal ice comes to rest uniformly in 3.52 s due to friction from the ice. What force does friction exert on the skater?

Animal dynamics. An adult 68 \(\mathrm{kg}\) cheetah can accelerate from rest to 20.1 \(\mathrm{m} / \mathrm{s}(45 \mathrm{mph})\) in 2.0 \(\mathrm{s}\) . Assuming constant acceleration, (a) find the net external force causing this acceleration. (b) Where does the force come from? That is, what exerts the force on the cheetah?

A hockey puck with mass 0.160 \(\mathrm{kg}\) is at rest on the horizontal, frictionless surface of a rink. A player applies a force of 0.250 \(\mathrm{N}\) to the puck, parallel to the surface of the ice, and continues to apply this force for 2.00 \(\mathrm{s}\) . What are the position and speed of the puck at the end of that time?

A dock worker applies a constant horizontal force of 80.0 \(\mathrm{N}\) to a block of ice on a smooth horizontal floor. The frictional force is negligible. The block starts from rest and moves 11.0 \(\mathrm{m}\) in the first 5.00 \(\mathrm{s}\) s. What is the mass of the block of ice?

(a) What is the mass of a book that weighs 3.20 \(\mathrm{N}\) in the laboratory? (b) In the same lab, what is the weight of a dog whose mass is 14.0 \(\mathrm{kg} ?\)

Superman throws a 2400 -N boulder at an adversary. What horizontal force must Superman apply to the boulder to give it a horizontal acceleration of 12.0 \(\mathrm{m} / \mathrm{s}^{2} ?\)

(a) How many newtons does a 150 lb person weigh? (b) Should a veterinarian be skeptical if someone said that her adult collie weighed 40 \(\mathrm{N} ?(\mathrm{c})\) Should a nurse question a medical chart which showed that an average-looking patient had a mass of 200 \(\mathrm{kg} ?\)

(a) An ordinary flea has a mass of 210\(\mu g .\) How many newtons does it weigh? (b) The mass of a typical froghopper is 12.3 \(\mathrm{mg} .\) How many newtons does it weigh (c) A house cat typically weighs 45 \(\mathrm{N}\) . How many pounds does it weigh and what is its mass in kilograms?

Calculate the mass (in SI units) of (a) a 160 -lb human being; (b) a \(1.9-\) -lb cockatoo. Calculate the weight (in English units) of \((c)\) a 2300 -kg rhinoceros; (d) a 22 -g sparrow.

An astronaut's pack weighs 17.5 \(\mathrm{N}\) when she is on earth but only 3.24 \(\mathrm{N}\) when she is at the surface of an asteroid. (a) What is the acceleration due to gravity on this asteroid? (b) What is the mass of the pack on the asteroid?

Interpreting a medical chart. You, a resident physician, are reading the medical chart of a normal adult female patient. Carelessly, one of the nurses has entered this woman's weight as a number without units. Another nurse has offered a suggestion for what the units might be. In each of the following cases, decide whether this nurse's suggestion is physically reasonable: (a) The number is \(150,\) and the nurse suggests that the units are kilograms. (b) The number is \(4.25,\) and the nurse suggests that the units are slugs. (c) The number is \(65,000,\) and the nurse suggests that the units are grams.

What does a 138 \(\mathrm{N}\) rock weigh if it is accelerating (a) upward at \(12 \mathrm{m} / \mathrm{s}^{2},\) (b) downward at 3.5 \(\mathrm{m} / \mathrm{s}^{2} ?\) (c) What would be the answers to parts (a) and (b) if the rock had a mass of 138 \(\mathrm{kg} ?(\mathrm{d})\) What would be the answers to parts (a) and (b) if the rock were moving with a constant upward velocity of 23 \(\mathrm{m} / \mathrm{s} ?\)

At the surface of Jupiter's moon Io, the acceleration due to gravity is 1.81 \(\mathrm{m} / \mathrm{s}^{2} .\) If a piece of ice weighs 44.0 \(\mathrm{N}\) at the surface of the earth, (a) what is its mass on the earth's surface? (b) What are its mass and weight on the surface of Io?

A scientific instrument that weighs 85.2 \(\mathrm{N}\) on the earth weighs 32.2 \(\mathrm{N}\) at the surface of Mercury. (a) What is the acceleration due to gravity on Mercury? (b) What is the instrument's mass on earth and on Mercury?

Planet \(\mathbf{X} !\) When venturing forth on Planet \(X,\) you throw a 5.24 kg rock upward at 13.0 \(\mathrm{m} / \mathrm{s}\) and find that it returns to the same level 1.51 \(\mathrm{s}\) later. What does the rock weigh on Planet \(\mathrm{X}\) ?

The driver of a 1750 \(\mathrm{kg}\) car traveling on a horizontal road at 110 \(\mathrm{km} / \mathrm{h}\) suddenly applies the brakes. Due to a slippery pavement, the friction of the road on the tires of the car, which is what slown the car, is 25\(\%\) of the weight of the car. (a) What is the acceleration of the car? (b) How many meters does it travel before stopping under these conditions?

You drag a heavy box along a rough horizontal floor by a horizontal rope. Identify the reaction force to each of the following forces: (a) the pull of the rope on the box, (b) the friction force on the box, (c) the normal force on the box, and (d) the weight of the box.

The upward normal force exerted by the floor is 620 \(\mathrm{N}\) on an elevator passenger who weighs 650 \(\mathrm{N} .\) What are the reaction forces to these two forces? Is the passenger accelerating? If so, what are the magnitude and direction of the acceleration?

A person throws a 2.5 lb stone into the air with an initial upward speed of 15 \(\mathrm{ft} / \mathrm{s}\) . Make a free-body diagram for this stone (a) after it is free of the person's hand and is traveling upward, (b) at its highest point, (c) when it is traveling downward, and (d) while it is being thrown upward, but is still in contact with the person's hand.

\(\bullet\) The driver of a car traveling at 65 mph suddenly hits his brakes on a horizontal highway.(a) Make a free-body diagram of the car while it is slowing down. (b) Make a free-body diagram of a passenger in a car that is accelerating on a freeway entrance ramp.

A tennis ball traveling horizontally at 22 \(\mathrm{m} / \mathrm{s}\) suddenly hits a vertical brick wall and bounces back with a horizontal velocity of 18 \(\mathrm{m} / \mathrm{s}\) . Make a free-body diagram of this ball (a) just before it hits the wall, (b) just after it has bounced free of the wall, and (c) while it is in contact with the wall.

Two crates, \(A\) and \(B,\) sit at rest side by side on a frictionless horizontal surface. The crates have masses \(m_{A}\) and \(m_{B} . A\) horizontal force \(\vec{\boldsymbol{F}}\) is applied to crate \(A\) and the two crates move off to the right. (a) Draw clearly labeled free-body diagrams for crate \(A\) and for crate \(B\) . Indicate which pairs of forces, if any, are third-law action- reaction pairs. (b) If the magnitude of force \(\vec{\boldsymbol{F}}\) is less than the total weight of the two crates, will it cause the crates to move? Explain.

A ball is hanging from a long string that is tied to the ceiling of a train car traveling eastward on horizontal tracks. An observer inside the train car sees the ball hang motionless. Draw a clearly labeled free-body diagram for the ball if (a) the train has a uniform velocity, and (b) the train is speeding up uniformly. Is the net force on the ball zero in either case? Explain.

A person drags her 65 \(\mathrm{N}\) suitcase along the rough horizontal floor by pulling upward at \(30^{\circ}\) above the horizontal with a 50 \(\mathrm{N}\) force. Make a free-body diagram of this suitcase.

A factory worker pushes horizontally on a 250 \(\mathrm{N}\) crate with a force of 75 \(\mathrm{N}\) on a horizontal rough floor. \(\mathrm{A} 135 \mathrm{N}\) crate rests on top of the one being pushed and moves along with it. Make a free-body diagram of each crate if the friction force exerted by the floor is less than the worker's push.

A uniform 25.0 \(\mathrm{kg}\) chain 2.00 \(\mathrm{m}\) long supports a 50.0 \(\mathrm{kg}\) chandelier in a large public building. Find the tension in (a) the bottom link of the chain, (b) the top link of the chain, and (c) the middle link of the chain.

An acrobat is hanging by his feet from a trapeze, while supporting with his hands a second acrobat who hangs below him. Draw separate free-body diagrams for the two acrobats.

A 275 \(\mathrm{N}\) bucket is lifted with an acceleration of 2.50 \(\mathrm{m} / \mathrm{s}^{2}\) by a 125 \(\mathrm{N}\) uniform vertical chain. Start each of the following parts with a free-body diagram. Find the tension in (a) the top link of the chain, (b) the bottom link of the chain, and (c) the middle link of the chain.

Human biomechanics. World-class sprinters can spring out of the starting blocks with an acceleration that is essentially horizontal and of magnitude 15 \(\mathrm{m} / \mathrm{s}^{2}\) . (a) How much horizontal force must a 55 -kg sprinter exert on the starting blocks during a start to produce this acceleration? (b) What exerts the force that propels the sprinter, the blocks or the sprinter himself?

A chair of mass 12.0 \(\mathrm{kg}\) is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force \(F=40.0 \mathrm{N}\) that is directed at an angle of \(37.0^{\circ}\) below the horizontal, and the chair slides along the floor. (a) Draw a clearly labeled free-body diagram for the chair. (b) Use your diagram and Newton's laws to calculate the normal force that the floor exerts on the chair.

Human biomechanics. The fastest pitched baseball was measured at 46 \(\mathrm{m} / \mathrm{s}\) . Typically, a baseball has a mass of 145 \(\mathrm{g}\) . If the pitcher exerted his force (assumed to be horizontal and constant) over a distance of \(1.0 \mathrm{m},(\mathrm{a})\) what force did he produce on the ball during this record-setting pitch? (b) Make free-body diagrams of the ball during the pitch and just after it has left the pitcher's hand.

You walk into an elevator, step onto a scale, and push the "up" button. You also recall that your normal weight is 625 \(\mathrm{N}\) . Start each of the following parts with a free-body diagram. (a) If the elevator has an acceleration of magnitude \(2.50 \mathrm{m} / \mathrm{s}^{2},\) what does the scale read? (b) If you start holding a 3.85 kg package by a light vertical string, what will be the tension in this string once the elevator begins accelerating?

A truck is pulling a car on a horizontal highway using a horizontal rope. The car is in neutral gear, so we can assume that there is no appreciable friction between its tires and the highway. As the truck is accelerating to highway speeds, draw a free-body diagram of (a) the car and (b) the truck. (c) What force accelerates this system forward? Explain how this force originates.

The space shuttle. During the first stage of its launch, a space shuttle goes from rest to 4973 \(\mathrm{km} / \mathrm{h}\) while rising a vertical distance of 45 \(\mathrm{km}\) . Assume constant acceleration and no variation in \(g\) over this distance. (a) What is the acceleration of the shuttle? (b) If a 55.0 \(\mathrm{kg}\) astronaut is standing on a scale inside the shuttle during this launch, how hard will the scale push on her? Start with a free-body diagram of the astronaut. (c) If this astronaut did not realize that the shuttle had left the launch pad, what would she think were her weight and mass?

A woman is standing in an elevator holding her 2.5 -kg briefcase by its handles. Draw a free-body diagram for the briefcase if the elevator is accelerating downward at 1.50 \(\mathrm{m} / \mathrm{s}^{2}\) and calculate the downward pull of the briefcase on the woman's arm while the elevator is accelerating.

An advertisement claims that a particular automobile can "stop on a dime." What net force would actually be necessary to stop a 850 -kg automobile traveling initially at 45.0 \(\mathrm{km} / \mathrm{h}\) in a distance equal to the diameter of a dime, which is 1.8 \(\mathrm{cm} ?\)

A rifle shoots a 4.20 g bullet out of its barrel. The bullet has a muzzle velocity of 965 \(\mathrm{m} / \mathrm{s}\) just as it leaves the barrel. Assuming a constant horizontal acceleration over a distance of 45.0 \(\mathrm{cm}\) starting from rest, with no friction between the bullet and the barrel, (a) what force does the rifle exert on the bullet while it is in the barrel? (b) Draw a free-body diagram of the bullet (i) while it is in the barrel and (ii) just after it has left the barrel. (c) How many \(g^{\prime}\) s of acceleration does the rifle give this bullet? (d) For how long a time is the bullet in the barrel?

A parachutist relies on air resistance (mainly on her parachute) to decrease her downward velocity. She and her parachute have a mass of 55.0 \(\mathrm{kg}\) , and at a particular moment air resistance exerts a total upward force of 620 \(\mathrm{N}\) on her and her parachute. (a) What is the weight of the parachutist? \((\mathrm{b})\) Draw a free-body diagram for the parachutist (see Section \(4.6 ) .\) Use that diagram to calculate the net force on the parachutist. Is the net force upward or downward? (c) What is the acceleration (magnitude and direction) of the parachutist?

A standing vertical jump. Basketball player Darrell Griffith is on record as attaining a standing vertical jump of 1.2 \(\mathrm{m}\) (4 ft). (This means that he moved upward by 1.2 \(\mathrm{m}\) after his feet left the floor.) Griffith weighed 890 \(\mathrm{N}(200 \mathrm{lb}) .\) (a) What was his speed as he left the floor? (b) If the time of the part of the jump before his feet left the floor was 0.300 s, what were the magnitude and direction of his acceleration (assuming it to be constant) while he was pushing against the floor? (c) Draw a free-body diagram of Griffith during the jump. (d) Use Newton's laws and the results of part (b) to calculate the force he applied to the ground during his jump.

You leave the doctor's office after your annual checkup and recall that you weighed 683 \(\mathrm{N}\) in her office. You then get into an elevator that, conveniently, has a scale. Find the magnitude and direction of the elevator's acceleration if the scale reads (a) \(725 \mathrm{N},(\mathrm{b}) 595 \mathrm{N}\)

Human biomechanics. The fastest served tennis ball, served by "Big Bill" Tilden in \(1931,\) was measured at 73.14 \(\mathrm{m} / \mathrm{s}\) . The mass of a tennis ball is \(57 \mathrm{g},\) and the ball is typically in contact with the tennis racquet for 30.0 \(\mathrm{ms}\) , with the ball starting from rest. Assuming constant acceleration, (a) what force did Big Bill's tennis racquet exert on the tennis ball if he hit it essentially horizontally? (b) Make free- body diagrams of the tennis ball during the serve and just after it has moved free of the racquet.

Extraterrestrial physics. You have landed on an unknown planet, Newtonia, and want to know what objects will weigh there. You find that when a certain tool is pushed on a frictionless horizontal surface by a 12.0 \(\mathrm{N}\) force, it moves 16.0 \(\mathrm{m}\) in the first 2.00 \(\mathrm{s}\) starting from rest. You next observe that if you release this tool from rest at 10.0 \(\mathrm{m}\) above the ground, it takes 2.58 s to reach the ground. What does the tool weigh on Newtonia and what would it weigh on Earth?

An athlete whose mass is 90.0 \(\mathrm{kg}\) is performing weight lifting exercises. Starting from the rest position, he lifts, with constant acceleration, a barbell that weighs 490 \(\mathrm{N} .\) He lifts the barbell a distance of 0.60 \(\mathrm{m}\) in 1.6 \(\mathrm{s}\) . (a) Draw a clearly labeled free-body force diagram for the barbell and for the athlete. (b) Use the diagrams in part (a) and Newton's laws to find the total force that the ground exerts on the athlete's feet as he lifts the barbell.

Jumping to the ground. A 75.0 \(\mathrm{kg}\) man steps off a platform 3.10 \(\mathrm{m}\) above the ground. He keeps his legs straight as he falls, but at the moment his feet touch the ground his knees begin to bend, and, treated as a particle, he moves an additional 0.60 \(\mathrm{m}\) before coming to rest. (a) What is his speed at the instant his feet touch the ground? (b) Treating him as a particle, what are the magnitude and direction of his acceleration as he slows down if the acceleration is constant? (c) Draw a free-body diagram of this man as he is slowing down. (d) Use Newton's laws and the results of part (b) to calculate the force the ground exerts on him while he is slowing down. Express this force in newtons and also as a multiple of the man's weight. (e) What are the magnitude and direction of the reaction force to the force you found in part (c)?

An electron \(\left(\mathrm{mass}=9.11 \times 10^{-31} \mathrm{kg}\right)\) leaves one end of a TV picture tube with zero initial speed and travels in a straight line to the accelerating grid, which is 1.80 \(\mathrm{cm}\) away. It reaches the grid with a speed of \(3.00 \times 10^{6} \mathrm{m} / \mathrm{s} .\) If the accelerating force is constant, compute (a) the acceleration of the electron, (b) the time it takes the electron to reach the grid, and (c) the net force that is accelerating the electron, in newtons. (You can ignore the gravitational force on the electron.)

BIO Bacterial motion. A bacterium using its flagellum as propulsion can move through liquids at a rate of 0.003 \(\mathrm{m} / \mathrm{s} .\) For a \(50 \mu \mathrm{m}-\) long bacterium, that is the equivalent of 60 cell lengths per second. Bacteria of that size have a mass of approximately \(1 \times 10^{-12} \mathrm{g}\). The viscous drag on a swimming bacterium is so great that if it stops beating its flagellum it will stop within a distance of 0.01 \(\mathrm{nm} .\) What is the acceleration that stops the bacterium? A. \(1.2 \times 10^{4} \mathrm{m} / \mathrm{s}^{2}\) B. \(5 \times 10^{5} \mathrm{m} / \mathrm{s}^{2}\) C. \(6 \times 10^{5} \mathrm{m} / \mathrm{s}^{2}\) D. \(9 \times 10^{5} \mathrm{m} / \mathrm{s}^{2}\)