Chapter 4

Two forces have the same magnitude \(F .\) What is the angle between the two vectors if their sum has a magnitude of (a) 2\(F ?\) (b) \(\sqrt{2} F ?(\mathrm{c})\) zero? Sketch the three vectors in each case.

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is \(60.0^{\circ} .\) If 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 \(A^{\prime}\) s rope.

Two forces, \(\overrightarrow{\boldsymbol{K}}_{1}\) and \(\overrightarrow{\boldsymbol{F}}_{2}\) act at a point. The magnitude of \(\overrightarrow{\boldsymbol{F}}_{1}\) is \(9.00 \mathrm{N},\) and its direction is \(60.0^{\circ}\) above the \(x\) -axis in the second quadrant. The magnitude of \(\overrightarrow{\boldsymbol{F}}_{2}\) is \(6.00 \mathrm{N},\) and its direction is \(53.1^{\circ}\) below the \(x\) -axis in the third quadrant. (a) What are the \(x\) - and \(y-\) components of the resultant force? (b) What is the magnitude of the resultant force?

If a net horizontal force of 132 \(\mathrm{N}\) is applied to a person with mass 60 \(\mathrm{kg}\) who is resting on the edge of a swimming pool, what horizontal acceleration is produced?

What magnitude of net force is required to give a \(135-\mathrm{kg}\) refrigerator an acceleration of magnitude 1.40 \(\mathrm{m} / \mathrm{s}^{2}\) ?

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?

A dockworker 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 5.00 \(\mathrm{s}\) . (a) What is the mass of the block of ice? (b) If the worker stops pushing at the end of 5.00 \(\mathrm{s}\) s, how far does the block move in the next 5.00 \(\mathrm{s?}\)

A hockey puck with mass 0.160 \(\mathrm{kg}\) is at rest at the origin \((x=0)\) on the horizontal, frictionless surface of the rink. At time \(t=0\) a player applies a force of 0.250 \(\mathrm{N}\) to the puck, parallel to the \(x\) -axis; he continues to apply this force until \(t=2.00 \mathrm{s}\) . (a) What are the position and speed of the puck at \(t=2.00 \mathrm{s?}\) (b) If the same force is again applied at \(t=5.00 \mathrm{s}\) , what are position and speed of the puck at \(t=7.00 \mathrm{s} ?\)

A crate with mass 32.5 \(\mathrm{kg}\) initially at rest on a warehouse floor is acted on by a net horizontal force of 140 \(\mathrm{N}\) . (a) What acceleration is produced? (b) How far does the crate travel in 10.0 \(\mathrm{s} ?\) (c) What is its speed at the end of 10.0 \(\mathrm{s} ?\)

A small \(8.00-\mathrm{kg}\) rocket burns fuel that exerts a time-varying upward force on the rocket. This force obeys the equation \(F=A+B t^{2} .\) Measurements show that at \(t=0,\) the force is \(100.0 \mathrm{N},\) and at the end of the first \(2.00 \mathrm{s},\) it is 150.0 \(\mathrm{N} .\) (a) Find the constants \(A\) and \(B,\) including their SI units. (b) Find the net force on this rocket and its acceleration (i) the instant after the fuel ignites and (ii) 3.00 s after fuel ignition. (c) Suppose you were using this rocket in outer space, far from all gravity. What would its acceleration be 3.00 s after fuel ignition?

An electron (mass \(=9.11 \times 10^{-31} \mathrm{kg} )\) 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 \(3.00 \times 10^{5} \mathrm{m} / \mathrm{s}\) . If the accelerating force is constant, compute (a) the acceleration; \((b)\) the time to reach the grid; (c) the net force, in newtons. (You can ignore the gravitational force on the electron.)

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 \(\mathrm{m} / \mathrm{s}^{2}\) ?

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

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?

World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude 15 \(\mathrm{m} / \mathrm{s}^{2}\) . How much horizontal force must a \(55-\mathrm{kg}\) sprinter exert on the starting blocks during a start to produce this acceleration? Which body exerts the force that propels the sprinter: the blocks or the sprinter herself?

Imagine that you are holding a book weighing 4 \(\mathrm{N}\) at rest on the palm of your hand. Complete the following sentences: (a) A downward force of magnitude 4 \(\mathrm{N}\) is exerted on the book by _____. (b) An upward force of magnitude _____ is exerted on _____ by your hand. (c) Is the upward force in part \((b)\) the reaction to the downward force in part \((a) ?\) (d) The reaction to the force in part (a) is a force of magnitude _____, exerted on _____ by _____. Its direction is _____. (e) The reaction to the force in part (b) is a force of magnitude _____, exerted on _____ by _____. Its direction is _____. (f) The forces in parts (a) and (b) are equal and opposite because of Newton's _____ law. (g) The forces in parts \((b)\) and (e) are equal and opposite because of Newton's _____ law. Now suppose that you exert an upward force of magnitude 5 \(\mathrm{N}\) on the book. (h) Does the book remain im equilibrium? (i) Is the force exerted on the book by your hand equal and opposite to the force exerted on the book by the earth? (i) Is the force exerted on the book by the earth equal and opposite to the force exerted on the earth by the book? (k) Is the force exerted on the book by your hand equal and opposite to the force exerted on your hand by the book? Finally, suppose you snatch your hand away while the book is moving upward. (I) How many forces then act on the book? (m) Is the book in equilibrium?

A bottle is given a push along a tabletop and slides off the edge of the table. Do not ignore air resistance. (a) What forces are exerted on the bottle while it is falling from the table to the floor? (b) What is the reaction to each force; that is, on which body and by which body is the reaction exerted?

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 student with mass 45 \(\mathrm{kg}\) jumps off a high diving board. Using \(6.0 \times 10^{24} \mathrm{kg}\) for the mass of the earth, what is the acceleration of the earth toward her as she accelerates toward the earth with an acceleration of 9.8 \(\mathrm{m} / \mathrm{s}^{2}\) ? Assume that the net force on the earth is the force of gravity she exerts on it.

An athlete throws a ball of mass \(m\) directly upward, and it feels no appreciable air resistance. Draw a free-body diagram of this ball while it is free of the athlete's hand and (a) moving upward; \((b)\) at its highest point; (c) moving downward. (d) Repeat parts \((a),(b),\) and \((c)\) if the athlete throws the ball at a \(60^{\circ}\) angle above the horizontal instead of directly upward.

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 \(\vec{F}\) 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 \(\vec{F}\) 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 castward 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 large box containing your new computer sits on the bed of your pickup truck. You are stopped at a red light. The light turns green and you stomp on the gas and the truck accelerates. To your horror, the box starts to slide toward the back of the truck. Draw clearly labeled free-body diagrams for the truck and for the box. Indicate pairs of forces, if any, that are third-law action- reaction pairs. (The bed of the truck is not frictionless.)

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.

A skier of mass 65.0 \(\mathrm{kg}\) is pulled up a snow-covered slope at constant speed by a tow rope that is parallel to the ground. The ground slopes upward at a constant angle of \(26.0^{\circ}\) above the horizontal, and you can ignore friction. (a) Draw a clearly labeled free-body diagram for the skier. (b) Calculate the tension in the tow rope.

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.

A. 22 rifle bullet, traveling at 350 \(\mathrm{m} / \mathrm{s}\) , strikes a large tree, which it penetrates to a depth of 0.130 \(\mathrm{m}\) . The mass of the bullet is 1.80 \(\mathrm{g}\) . Assume a constant retarding force. (a) How much time is required for the bullet to stop? (b) What force, in newtons, does the tree exert on the bullet?

You have just landed on Planet \(X\) . You take out a 100 -g ball, release it from rest from a height of \(10.0 \mathrm{m},\) and measure that it takes 2.2 \(\mathrm{s}\) to reach the ground. You can ignore any force on the ball from the atmosphere of the planet. How much does the \(100-\mathrm{g}\) ball weigh on the surface of Planet \(\mathrm{X}\) ?

An oil tanker's engines have broken down, and the wind is blowing the tanker straight toward a recf at a constant spocd of 1.5 \(\mathrm{m} / \mathrm{s}(\text { Fig. } 4.37) .\) When the tanker is 500 \(\mathrm{m}\) m from the reef, the wind dies down just as the engineer gets the engines going again. The rudder is stuck, so the only choice is to try to accelerate straight backward away from the reef. The mass of the tanker and cargo is \(3.6 \times 10^{7} \mathrm{kg}\) , and the engines produce a net horizontal force of \(8.0 \times 10^{4} \mathrm{N}\) on the tanker. Will the ship hit the reef? If it does, will the oil be safe? The hull can withstand an impact at a speed of 0.2 \(\mathrm{m} / \mathrm{s}\) or less. You can ignore the retarding force of the water on the tanker's hull.

An advertisement claims that a particular automobile can "stop on a dime." What net force would actually be necessary to stop a \(850-\mathrm{kg}\) 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 \(4.80-\mathrm{kg}\) bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 75.0 \(\mathrm{N}\) . (a) Draw the free-body force diagram for the bucket. In terms of the forces on your diagram, what is the net force on the bucket? (b) Apply Newton's second law to the bucket and find the maximum upward acceleration that can be given to the bucket without breaking the cord.

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 air resistance exerts a total upward force of 620 \(\mathrm{N}\) on her and her parachute. (a) What is the weight of the parachutist? (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?

An astronaut is tethered by a strong cable to a spacecraft. The astronaut and her spacesuit have a total mass of 105 \(\mathrm{kg}\) , while the mass of the cable is negligible. The mass of the spacecraft is \(9.05 \times 10^{4} \mathrm{kg}\) . The spacecraft is from any large astronomical bodies, so we can ignore the gravitational forces on it and the astronaut. We also assume that both the spacecraft and the astronaut are initially at rest in an inertial reference frame. The astronaut then pulls on the cable with a force of 80.0 \(\mathrm{N}\) . (a) What force bodies, so we can ignore the gravitational forces on it and the astronaut. We also assume that both the spacecraft and the astronaut are initially at rest in an inertial reference frame. The astronaut then pulls on the cable with a force of 80.0 \(\mathrm{N}\) . (a) What force does the cable exert on the astronaut? (b) Since \(\Sigma \overrightarrow{\boldsymbol{F}}=m \overrightarrow{\mathbf{a}},\) how can a "massless" \((m=0)\) cable exert a force? (c) What is the astronaut's acceleration? (d) What force does the cable exert on the spacecraft? (e) What is the acceleration of the spacecraft?

To study damage to aircraft that collide with large birds, you design a test gun that will accelerate chicken-sized objects so that their displacement along the gun barrel is given by \(x=\) \(\left(9.0 \times 10^{3} \mathrm{m} / \mathrm{s}^{2}\right) t^{2}-\left(8.0 \times 10^{4} \mathrm{m} / \mathrm{s}^{3}\right) t^{3} .\) The object leaves the end of the barrel at \(t=0.025 \mathrm{s}\) (a) How long must the gun barrel be? \((b)\) What will be the speed of the objects as they leave the end of the barrel? (c) What net force must be exerted on a \(1.50-k g\) object at \((\mathrm{i}) t=0\) and (ii) \(t=0.025 \mathrm{s} ?\)

A spacecraft descends vertically near the surface of Planet X. An upward thrust of 25.0 \(\mathrm{kN}\) from its engines slows it down at a rate of \(1.20 \mathrm{m} / \mathrm{s}^{2},\) but it speeds up at a rate of 0.80 \(\mathrm{m} / \mathrm{s}^{2}\) with an upward thrust of 10.0 \(\mathrm{kN} .(\mathrm{a})\) In each case, what is the direction of the acceleration of the spacecraft? (b) Draw a free-body diagram for the spacecraft. In each case, speeding up or slowing down, what is the direction of the net force on the spacecraft? (c) Apply Newton's second law to each case, slowing down or speeding up, and use this to find the spacecraft's weight near the surface of Planet X.

A 6.50-kg instrument is hanging by a vertical wire inside a space ship that is blasting off at the surface of the earth. This ship starts from rest and reaches an altitude of 276 \(\mathrm{m}\) in 15.0 s with constant acceleration. (a) Draw a free-body diagram for the instrument during this time. Indicate which force is greater. (b) Find the force that the wire exerts on the instrument.

A gymnast of mass \(m\) climbs a vertical rope attached to the ceiling. You can ignore the weight of the rope. Draw a free-body diagram for the gymnast. Calculate the tension in the rope if the gymnast (a) climbs at a constant rate; (b) hangs motionless on the rope; (c) accelerates up the rope with an acceleration of magnitude \(|\vec{a}| ;(\text { d) slides down the rope with a downward acceleration of }\) magnitude \(|\vec{a}| .\)

A loaded elevator with very worn cables has a total mass of 2200 \(\mathrm{kg}\) , and the cables can withstand a maximum tension of \(28,000 \mathrm{N} .\) (a) Draw the free-body force diagram for the elevator. In terms of the forces on your diagram, what is the net force on the elevator? Apply Newton's second law to the elevator and find the maximum upward acceleration for the elevator if the cables are not to break. (b) What would be the answer to part (a) if the elevator were on the moon, where \(g=1.62 \mathrm{m} / \mathrm{s}^{2} ?\)

Jumping to the Ground. A 75.0 -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 is his acceleration (magnitude and direction) as he slows down, if the acceleration is assumed to be constant? (c) Draw his free-body diagram (see Section 4.6 ). In terms of the forces on the diagram, what is the net force on him? Use Newton's laws and the results of part \((b)\) to calculate the average force his feet exert on the ground while he slows down. Express this force in newtons and also as a multiple of his weight.

A \(4.9-\mathrm{N}\) hammer head is stopped from an initial downward velucity of 3.2 \(\mathrm{m} / \mathrm{s}\) in a distance of 0.45 \(\mathrm{cm}\) by a nail in a pine board. In addition to its weight, there is a \(15-\mathrm{N}\) downward force on the hammer head applied by the person using the hammer. Assume that the acceleration of the hammer head is constant while it is in contact with the nail and moving downward. (a) Draw a free-body diagram for the hammer head. Identify the reaction force to each action force in the diagram. (b) Calculate the downward force \(\overrightarrow{\boldsymbol{F}}\) exerted by the hammer head on the nail while the hammer head is in contact with the nail and moving downward. (c) Suppose the nail is in hardwood and the distance the hammer head travels in coming to rest is only 0.12 \(\mathrm{cm} .\) The downward forces on the hammer head are the same as on part (b). What then is the force \(\overrightarrow{\boldsymbol{F}}\) exerted by the hammer head on the nail while the hammer head is in contact with the nail and moving downward?

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 his feet exert on the ground as he lifts the barbell.

A hot-air balloon consists of a basket, one passenger, and some cargo. Let the total mass be \(M\) . Even though there is an upward lift force on the balloon, the balloon is initially accelerating downward at a rate of \(g / 3\) . (a) Draw a free-body diagram for the descending balloon. (b) Find the upward lift force in terms of the initial total weight Mg. (c) The passenger notices that he is heading straight for a waterfall and decides he needs to go up. What fraction of the total weight must he drop overboard so that the balloon accelerates upward at a rate of \(g / 2 ?\) Assume that the upward lift force remains the same.

A student tries to raise a chain consisting of three identical links. Each link has a mass of 300 \(\mathrm{g}\) . The three-piece chain is connected to a string and then suspended vertically, with the student holding the upper end of the string and pulling upward. Because of the student's pull, an upward force of 12 \(\mathrm{N}\) is applied to the chain by the string. (a) Draw a free-body diagram for each of the links in the chain and also for the entire chain considered as a single body. (b) Use the results of part (a) and Newton's laws to find ( ) the acceleration of the chain and (ii) the force exerted by the top link on the middle link.

The position of a \(2.75 \times 10^{5} \mathrm{N}\) training helicopter under test is given by \(\vec{r}=\left(0.020 \mathrm{m} / \mathrm{s}^{3}\right) t^{3} \hat{\imath}+(2.2 \mathrm{m} / \mathrm{s}) f \hat{\jmath}-\left(0.060 \mathrm{m} / \mathrm{s}^{2}\right) t^{2} \hat{k}\) Find the net force on the helicopter at \(t=5.0 \mathrm{s}\)

An object with mass \(m\) moves along the \(x\) -axis. Its position as a function of time is given by \(x(t)=A t-B t^{3},\) where \(A\) and \(B\) are constants. Calculate the net force on the object as a function of time.

An object with mass \(m\) initially at rest is acted on by a force \(\vec{F}=k_{1} \hat{z}+k_{2} t^{3}\) , where \(k_{1}\) and \(k_{2}\) are constants. Calculate the velocity \(\vec{v}(t)\) of the object as a function of time.

If we know \(F(t),\) the force as a function of time, for straight-line motion, Newton's second law gives us \(a(t),\) the acceleration as a function of time. We can then integrate \(a(t)\) to find \(v(t)\) and \(x(t)\) . However, suppose we know \(F(v)\) instead. (a) The net force on a body moving along the \(x\) -axis equals \(-C v^{2} .\) Use Newton's second law written as \(\Sigma F=m d v / d t\) and two integrations to show that \(x-x_{0}=(m / C) \ln \left(v_{0} / v\right) .\) (b) Show that Newton's second law can be written as \(\Sigma F=m v d v / d x .\) Derive the same expression as in part (a) using this form of the second law and one integration.

An object of mass \(m\) is at rest in equilibrium at the origin. At \(t=0\) a new force \(\vec{F}(t)\) is applied that has components $$ F_{x}(t)=k_{1}+k_{2} y \quad F_{y}(t)=k_{3} t $$ where \(k_{1}, k_{2},\) and \(k_{3}\) are constants. Calculate the position \(\vec{r}(t)\) and velocity \(\vec{v}(t)\) vectors as functions of time.