Chapter 5: Newton's Laws of MotionContent check — University Physics 1, Chapter 5: Newton's Laws of Motion

Find the magnitude and direction of the net force in the ice skater example just given.

A skydiver opens their parachute, and shortly thereafter, they are moving at constant velocity. (a) What forces are acting on them? (b) Which force is bigger?

At the time of its launch, the HMS Titanic was the most massive mobile object ever built, with a mass of $6.0\times {10}^7\text{kg}$. If a force of 6 MN $(6\times {10}^6\text{N})$ was applied to the ship, what acceleration would it experience?

A 550-kg sports car collides with a 2200-kg truck, and during the collision, the net force on each vehicle is the force exerted by the other. If the magnitude of the truck’s acceleration is $10{\text{m/s}}^2,$ what is the magnitude of the sports car’s acceleration?

A car has forces acting on it, as shown below. The mass of the car is 1000.0 kg. The road is slick, so friction can be ignored. (a) What is the net force on the car? (b) What is the acceleration of the car?

For Example 5.8, find the acceleration when the farmer’s applied force is 230.0 N.

Two blocks are at rest and in contact on a frictionless surface as shown below, with $m_1=2.0\text{kg},$ $m_2=6.0\text{kg},$ and applied force 24 N. (a) Find the acceleration of the system of blocks. (b) Suppose that the blocks are later separated. What force will give the second block, with the mass of 6.0 kg, the same acceleration as the system of blocks?

A force of 1150 N acts parallel to a ramp to push a 250-kg gun safe into a moving van. The ramp is frictionless and inclined at $17\text{°}.$ (a) What is the acceleration of the safe up the ramp? (b) If we consider friction in this problem, with a friction force of 120 N, what is the acceleration of the safe?

One end of a 3.0-m rope is tied to a tree; the other end is tied to a car stuck in the mud. The motorist pulls sideways on the midpoint of the rope, displacing it a distance of 0.25 m. If he exerts a force of 200.0 N under these conditions, determine the force exerted on the car.

(a) Draw the free-body diagram for the situation shown. (b) Redraw it showing components; use x-axes parallel to the two ramps.

What properties do forces have that allow us to classify them as vectors?

A woman was transporting an open box of cupcakes to a school party. The car in front of her stopped suddenly; she applied her brakes immediately. She was wearing her seat belt and suffered no physical harm (just a great deal of embarrassment), but the cupcakes flew into the dashboard and became “smushcakes.” Explain what happened.

A rock is thrown straight up. At the top of the trajectory, the velocity is momentarily zero. Does this imply that the force acting on the object is zero? Explain your answer.

How much does a 70-kg astronaut weight in space, far from any celestial body? What is her mass at this location?

When you stand on Earth, your feet push against it with a force equal to your weight. Why doesn’t Earth accelerate away from you?

Identify the action and reaction forces in the following situations: (a) Earth attracts the Moon, (b) a boy kicks a football, (c) a rocket accelerates upward, (d) a car accelerates forward, (e) a high jumper leaps, and (f) a bullet is shot from a gun.

(a) Why does an ordinary rifle recoil (kick backward) when fired? (b) The barrel of a recoilless rifle is open at both ends. Describe how Newton’s third law applies when one is fired. (c) Can you safely stand close behind one when it is fired?

A particle is moving to the right. (a) Can the force on it be acting to the left? If yes, what would happen? (b) Can that force be acting downward? If yes, why?

If a book is located on a table, how many forces should be shown in a free-body diagram of the book? Describe them.

Two ropes are attached to a tree, and forces of ${\overrightarrow{F}}_1=$ $2.0\widehat{i}+$ $4.0\widehat{j}\text{N}$ and ${\overrightarrow{F}}_2=$ $3.0\widehat{i}+$ $6.0\widehat{j}\text{N}$ are applied. The forces are coplanar (in the same plane). (a) What is the resultant (net force) of these two force vectors? (b) Find the magnitude and direction of this net force.

Two teenagers are pulling on ropes attached to a tree. The angle between the ropes is $30.0\text{°}$. David pulls with a force of 400.0 N and Stephanie pulls with a force of 300.0 N. (a) Find the component form of the net force. (b) Find the magnitude of the resultant (net) force on the tree and the angle it makes with David’s rope.

While sliding a couch across a floor, Andrea and Jennifer exert forces ${\overrightarrow{F}}_{\text{A}}$ and ${\overrightarrow{F}}_{\text{J}}$ on the couch. Andrea’s force is due north with a magnitude of 130.0 N and Jennifer’s force is $32\text{°}$ east of north with a magnitude of 180.0 N. (a) Find the net force in component form. (b) Find the magnitude and direction of the net force. (c) If Andrea and Jennifer’s housemates, David and Stephanie, disagree with the move and want to prevent its relocation, with what combined force ${\overrightarrow{F}}_{\text{DS}}$ should they push so that the couch does not move?

If the sprinter from the previous problem accelerates at that rate for 20.00 m and then maintains that velocity for the remainder of a 100.00-m dash, what will her time be for the race?

Astronauts in orbit are apparently weightless. This means that a clever method of measuring the mass of astronauts is needed to monitor their mass gains or losses, and adjust their diet. One way to do this is to exert a known force on an astronaut and measure the acceleration produced. Suppose a net external force of 50.0 N is exerted, and an astronaut’s acceleration is measured to be $0.893{\text{m/s}}^2$. (a) Calculate her mass. (b) By exerting a force on the astronaut, the vehicle in which she orbits experiences an equal and opposite force. Use this knowledge to find equations for the acceleration of the spaceship and for the apparent acceleration of the spaceship and for the apparent acceleration that would be measured by an observer at rest relative to the spaceship. (c) Discuss how this would affect the measurement of the astronaut’s acceleration. Propose a method by which recoil of the vehicle is avoided.

The rocket sled shown below accelerates opposite to the motion at a rate of $196{\text{m/s}}^2$. What force is necessary to produce this acceleration opposite to the motion? Assume that the rockets are off. The mass of the system is $2.10\times {10}^3$ kg.

What is the acceleration opposite to the motion of the rocket sled if it comes to rest in 1.10 s from a speed of 1000.0 km/h? (Such acceleration opposite to the motion caused one test subject to black out and have temporary blindness.)

A powerful motorcycle can produce an acceleration of $3.50{\text{m/s}}^2$ while traveling at 90.0 km/h. At that speed, the forces resisting motion, including friction and air resistance, total 400.0 N. (Air resistance is analogous to air friction. It always opposes the motion of an object.) What is the magnitude of the force that motorcycle exerts backward on the ground to produce its acceleration if the mass of the motorcycle with rider is 245 kg?

The driver in the previous problem applies the brakes when the car is moving at 90.0 km/h, and the car comes to rest after traveling 40.0 m. What is the net force on the car during its acceleration opposite to the motion?

A particle of mass 2.0 kg is acted on by a single force ${\overrightarrow{F}}_1=18\widehat{i}\text{N}.$ (a) What is the particle’s acceleration? (b) If the particle starts at rest, how far does it travel in the first 5.0 s?

Find the acceleration of the body of mass 5.0 kg shown below.

The weight of an astronaut plus their space suit on the Moon is only 250 N. (a) How much does the suited astronaut weigh on Earth? (b) What is the mass on the Moon? On Earth?

A rocket sled accelerates at a rate of ${49.0\text{m/s}}^2$. Its passenger has a mass of 75.0 kg. (a) Calculate the horizontal component of the force the seat exerts against his body. Compare this with his weight using a ratio. (b) Calculate the direction and magnitude of the total force the seat exerts against his body.

A body of mass 2.00 kg is pushed straight upward by a 25.0 N vertical force. What is its acceleration?

A body with a mass of 10.0 kg is assumed to be in Earth’s gravitational field with $g=9.80{\text{m/s}}^2$. What is the net force on the body if there are no other external forces acting on the object?

A baseball catcher is performing a stunt for a television commercial. He will catch a baseball (mass 145 g) dropped from a height of 60.0 m above his glove. His glove stops the ball in 0.0100 s. What is the force exerted by his glove on the ball?

(a) What net external force is exerted on a 1100.0-kg artillery shell fired from a battleship if the shell is accelerated at $2.40\times {10}^4{\text{m/s}}^2?$ (b) What is the magnitude of the force exerted on the ship by the artillery shell, and why?

A history book is lying on top of a physics book on a desk, as shown below; a free-body diagram is also shown. The history and physics books weigh 14 N and 18 N, respectively. Identify each force on each book with a double subscript notation (for instance, the contact force of the history book pressing against physics book can be described as ${\overrightarrow{F}}_{\text{HP}}$), and determine the value of each of these forces, explaining the process used.

A leg is suspended in a traction system, as shown below. (a) Which pulley in the figure is used to calculate the force exerted on the foot? (b) What is the tension in the rope? Here $\overrightarrow{T}$ is the tension, ${\overrightarrow{w}}_{\text{leg}}$ is the weight of the leg, and $\overrightarrow{w}$ is the weight of the load that provides the tension. Express answers in terms of the unknown masses and angle.

Two teams of nine members each engage in a tug-of-war, pulling in opposite directions on a horizontal rope. Each of the first team’s members has an average mass of 68 kg and exerts an average force of 1350 N horizontally on the ground as they pull on the rope. Each of the second team’s members has an average mass of 73 kg and exerts an average force of 1365 N horizontally on the ground as they pull on the rope in the opposite direction. (a) What is the magnitude of the acceleration of the two teams, and which team wins? (b) What is the tension in the section of rope between the teams?

(a) Calculate the tension in a vertical strand of spider web if a spider of mass $2.00\times {10}^{\mathrm{-5}}\text{kg}$ hangs motionless on it. (b) Calculate the tension in a horizontal strand of spider web if the same spider sits motionless in the middle of it much like the tightrope walker in Figure 5.26. The strand sags at an angle of $12\text{°}$ below the horizontal. Compare this with the tension in the vertical strand (find their ratio).

Show that, as explained in the text, a force $F_{\perp }$ exerted on a flexible medium at its center and perpendicular to its length (such as on the tightrope wire in Figure 5.26) gives rise to a tension of magnitude $T=$ $F_{\perp }\text{/}2\text{sin}\left(\theta \right)$.

A bird has a mass of 26 g and perches in the middle of a stretched telephone line. (a) Show that the tension in the line can be calculated using the equation $T=\frac{mg}{2\text{sin}\theta }$. Determine the tension when (b) $\theta =5\text{°}$ and (c) $\theta =0.5\text{°}$. Assume that each half of the line is straight.

Consider the baby being weighed in the following figure. (a) What is the mass of the infant and basket if a scale reading of 55 N is observed? (b) What is tension $T_1$ in the cord attaching the baby to the scale? (c) What is tension $T_2$ in the cord attaching the scale to the ceiling, if the scale has a mass of 0.500 kg? (d) Sketch the situation, indicating the system of interest used to solve each part. The masses of the cords are negligible.

A 2.0-kg block is on a perfectly smooth ramp that makes an angle of $30\text{°}$ with the horizontal. (a) What is the block’s acceleration down the ramp and the force of the ramp on the block? (b) What force applied upward along and parallel to the ramp would allow the block to move with constant velocity?

A car moves along a horizontal road. Draw a free-body diagram; be sure to include the friction of the road that opposes the forward motion of the car.

The traffic light hangs from the cables as shown. Draw a free-body diagram on a coordinate plane for this situation.

Two forces of 25 and 45 N act on an object. Their directions differ by $70\text{°}$. The resulting acceleration has magnitude of $10.0{\text{m/s}}^2.$ What is the mass of the body?

Draw a free-body diagram of a diver who has entered the water, moved downward, and is acted on by an upward force due to the water which balances the weight (that is, the diver is suspended).

(a) Find an equation to determine the magnitude of the net force required to stop a car of mass m, given that the initial speed of the car is $v_0$ and the stopping distance is x. (b) Find the magnitude of the net force if the mass of the car is 1050 kg, the initial speed is 40.0 km/h, and the stopping distance is 25.0 m.

Find the acceleration of the body of mass 10.0 kg shown below.

Force ${\overrightarrow{F}}_{\text{B}}$ has twice the magnitude of force ${\overrightarrow{F}}_{\text{A}}.$ Find the direction in which the particle accelerates in this figure.

A force acts on a car of mass m so that the speed v of the car increases with position x as $v=k{x}^2$, where k is constant and all quantities are in SI units. Find the force acting on the car as a function of position.

Two boxes, A and B, are at rest. Box A is on level ground, while box B rests on an inclined plane tilted at angle $\theta$ with the horizontal. (a) Write expressions for the normal force acting on each block. (b) Compare the two forces; that is, tell which one is larger or whether they are equal in magnitude. (c) If the angle of incline is $10\text{°}$, which force is greater?

As shown below, two identical springs, each with the spring constant 20 N/m, support a 15.0-N weight. (a) What is the tension in spring A? (b) What is the amount of stretch of spring A from the rest position?

In building a house, carpenters use nails from a large box. The box is suspended from a spring twice during the day to measure the usage of nails. At the beginning of the day, the spring stretches 50 cm. At the end of the day, the spring stretches 30 cm. What fraction or percentage of the nails have been used?

Two forces are applied to a 5.0-kg object, and it accelerates at a rate of $2.0{\text{m/s}}^2$ in the positive y-direction. If one of the forces acts in the positive x-direction with magnitude 12.0 N, find the magnitude of the other force.

If two tugboats pull on a disabled vessel, as shown here in an overhead view, the disabled vessel will be pulled along the direction indicated by the result of the exerted forces. (a) Draw a free-body diagram for the vessel. Assume no friction or drag forces affect the vessel. (b) Did you include all forces in the overhead view in your free-body diagram? Why or why not?

On June 25, 1983, shot-putter Udo Beyer of East Germany threw the 7.26-kg shot 22.22 m, which at that time was a world record. (a) If the shot was released at a height of 2.20 m with a projection angle of $45.0\text{°}$, what was its initial velocity? (b) If while in Beyer’s hand the shot was accelerated uniformly over a distance of 1.20 m, what was the net force on it?

A body of mass m has initial velocity $v_0$ in the positive x-direction. It is acted on by a constant force F for time t until the velocity becomes zero; the force continues to act on the body until its velocity becomes $\text{−}{v}_0$ in the same amount of time. Write an expression for the total distance the body travels in terms of the variables indicated.

A 120-kg astronaut is riding in a rocket sled that is sliding along an inclined plane. The sled has a horizontal component of acceleration of $5.0\text{m}\text{/}{\text{s}}^2$ and a downward component of $3.8\text{m}\text{/}{\text{s}}^2$. Calculate the magnitude of the force on the rider by the sled. (Hint: Remember that gravitational acceleration must be considered.)

Suppose that you are viewing a soccer game from a helicopter above the playing field. Two soccer players simultaneously kick a stationary soccer ball on the flat field; the soccer ball has mass 0.420 kg. The first player kicks with force 162 N at $9.0\text{°}$ north of west. At the same instant, the second player kicks with force 215 N at $15\text{°}$ east of south. Find the acceleration of the ball in $\widehat{i}$ and $\widehat{j}$ form.

A 0.0502-kg pair of fuzzy dice is attached to the rearview mirror of a car by a short string. The car accelerates at constant rate, and the dice hang at an angle of $3.20\text{°}$ from the vertical because of the car’s acceleration. What is the magnitude of the acceleration of the car?

Hanging from the ceiling over a baby bed, well out of baby’s reach, is a string with plastic shapes, as shown here. The string is taut (there is no slack), as shown by the straight segments. Each plastic shape has the same mass m, and they are equally spaced by a distance d, as shown. The angles labeled $\theta$ describe the angle formed by the end of the string and the ceiling at each end. The center length of sting is horizontal. The remaining two segments each form an angle with the horizontal, labeled $\varphi$. Let $T_1$ be the tension in the leftmost section of the string, $T_2$ be the tension in the section adjacent to it, and $T_3$ be the tension in the horizontal segment. (a) Find an equation for the tension in each section of the string in terms of the variables m, g, and $\theta$. (b) Find the angle $\varphi$ in terms of the angle $\theta$. (c) If $\theta =5.10\text{°}$, what is the value of $\varphi$? (d) Find the distance x between the endpoints in terms of d and $\theta$.

An object is acted on by three simultaneous forces: ${\overrightarrow{F}}_1=$ $(\mathrm{-3.00}\widehat{i}+2.00\widehat{j})\text{N}$, ${\overrightarrow{F}}_2=$ $(6.00\widehat{i}-4.00\widehat{j})\text{N}$, and ${\overrightarrow{F}}_3=$ $(2.00\widehat{i}+5.00\widehat{j})\text{N}$. The object experiences acceleration of $4.23{\text{m/s}}^2$. (a) Find the acceleration vector in terms of m. (b) Find the mass of the object. (c) If the object begins from rest, find its speed after 5.00 s. (d) Find the components of the velocity of the object after 5.00 s.

A drone is being directed across a frictionless ice-covered lake. The mass of the drone is 1.50 kg, and its velocity is $3.00\widehat{i}\text{m}\text{/}\text{s}$. After 10.0 s, the velocity is $9.00\widehat{i}+$ $4.00\widehat{j}\text{m}\text{/}\text{s}$. If a constant force in the horizontal direction is causing this change in motion, find (a) the components of the force and (b) the magnitude of the force.