Mechanics

A person can dive into the water from a height of 10 m without injury, but a person who jumps off the roof of a 10-m-tall building and lands on a concrete street is likely to be seriously injured. Why is there a difference?

Why are cars designed to crumple in front and back for safety? Why not for side collisions and rollovers?

When a string is barely strong enough lifts a heavy weight, it can lift the weight by a steady pull; but if you jerk the string, it will break. Explain in terms of Newton’s laws of motion.

A large crate is suspended from the end of a vertical rope. Is the tension in the rope greater when the crate is at rest or when it is moving upward at constant speed? If the crate is traveling upward, is the tension in the rope greater when the crate is speeding up or when it is slowing down? In each case, explain in terms of Newton’s laws of motion.

Which feels a greater pull due to the earth’s gravity: a 10-kg stone or a 20-kg stone? If you drop the two stones, why doesn’t the 20-kg stone fall with twice the acceleration of the 10-kg stone? Explain.

Why is it incorrect to say that 1.0 kg equals 2.2 lb?

A horse is hitched to a wagon. Since the wagon pulls back on the horse just as hard as the horse pulls on the wagon, why doesn’t the wagon remain in equilibrium, no matter how hard the horse pulls?

True or false? You exert a push P on an object, and it pushes back on you with a force F. If the object is moving at a constant velocity, then F is equal to P, but if the object is being accelerated, then P must be greater than F.

A large truck and a small compact car have a head-on collision. During the collision, the truck exerts a force ${\overrightarrow{\mathbf{F}}}_{\mathbf{T}\mathbf{ }\mathbf{on}\mathbf{ }\mathbf{C}}$ on the car, and the car exerts a force ${\overrightarrow{\mathbf{F}}}_{\mathbf{C}\mathbf{ }\mathbf{on}\mathbf{ }\mathbf{T}}$ on the truck. Which force has the larger magnitude, or are they the same? Does your answer depend on how fast each vehicle was moving before the collision? Why or why not?

When a car comes to a stop on a level highway, what force causes it to slow down? When the car increases its speed on the same highway, what force causes it to speed up? Explain.

A small compact car is pushing a large van that has broken down, and they travel along the road with equal velocities and accelerations. While the car is speeding up, is the force it exerts on the van larger than, smaller than, or the same magnitude as the force the van exerts on it? Which vehicle has the larger net force on it, or are the net forces the same? Explain.

Consider a tug-of-war between two people who pull in opposite directions on the ends of a rope. By Newton’s third law, the force that A exerts on B is just as great as the force that B exerts on A. So what determines who wins? (Hint: Draw a free-body diagram showing all the forces that act on each person.)

Boxes A and B are in contact on a horizontal, frictionless surface. You push on box A with a horizontal 100-N force (Fig. Q4.31). Box A weighs 150 N, and box B weighs 50 N. Is the force that box A exerts on box B equal to 100 N, greater than 100 N, or less than 100 N? Explain.

Suppose you are in a rocket with no windows, traveling in deep space far from other objects. Without looking outside the rocket or making any contact with the outside world, explain how you could determine whether the rocket is 

(a) moving forward at a constant  80% of the speed of light and 

(b) accelerating in the forward direction.

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is  $60°$. If Rover exerts a force of 270 N and Fido exerts a force of 300 N, find the magnitude of the resultant force and the angle it makes with Rover’s rope.

To extricate an SUV stuck in the mud, workmen use three horizontal ropes, producing the force vectors shown in Fig. E4.2.

(a) Find the x- and y-components of each of the three pulls.

(b) Use the components to find the magnitude and direction of the resultant of the three pulls.

bio Jaw Injury. Due to a jaw injury, a patient must wear a strap (Fig. E4.3) that produces a net upward force of 5.00 N on his chin. The tension is the same throughout the strap. To what tension must the strap be adjusted to provide the necessary upward force?

A man is dragging a trunk up the loading ramp of a mover’s truck. The ramp has a slope angle of  $\mathbf{20}\mathbf{°}$, and the man pulls upward with a force  $\overrightarrow{\mathbf{F}}$whose direction makes an angle of  $\mathbf{30}\mathbf{°}$ with the ramp (Fig. E4.4).

 (a) How large a force  $\overrightarrow{\mathit{F}}$ is necessary for the component  ${\mathit{F}}_{\mathbf{x}}$ parallel to the ramp to be 90.0 N?  

(b) How large will the component ${\mathit{F}}_{\mathbf{y}}$  perpendicular to the ramp be then?

Forces ${\overrightarrow{F}}_1$  and ${\overrightarrow{F}}_2$  act at a point. The magnitude of  ${\overrightarrow{F}}_1$ is 9.00 N, and its direction is ${60}^{\circ }$  above the x-axis in the second quadrant. The magnitude of   ${\overrightarrow{F}}_2$is 6.00 N, and its direction is   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?

An electron (mass = $\mathbf{9}\mathbf{.}\mathbf{11}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{31}}\mathit{k}\mathit{g}$) 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 cm away. It reaches the grid with a speed of $\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathit{m}\mathbf{/}\mathit{s}$ . If the accelerating force is constant, compute

(a) the acceleration; 

(b) the time to reach the grid; and 

(c) the net force, in newtons. Ignore the gravitational force on the electron.

A  68.5 kg skater moving initially at  2.4 m/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?

You walk into an elevator, step onto a scale, and push the “up” button. You recall that your normal weight is 625 N. Draw a free-body diagram. 

(a) When the elevator has an upward acceleration of magnitude $2.5 m/S^2$, what does the scale read? 

(b) If you hold a 3.85 kg package by a light vertical string, what will be the tension in this string when the elevator accelerates as in part (a)?

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 N to the box and produces an acceleration of magnitude $\mathbf{2}\mathbf{.}\mathbf{2}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$, what is the mass of the box?

A dockworker applies a constant horizontal force of 80.0 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 m in 5.00 s. 

(a) What is the mass of the block of ice? 

(b) If the worker stops pushing at the end of 5.00 s, how far does the block move in the next 5.00 s?

A hockey puck with mass 0.160 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 N to the puck, parallel to the x-axis; she continues to apply this force until t = 2.00 s.

(a) What are the position and speed of the puck at t = 2.00 s?

(b) If the same force is again applied at t = 5.00 s, what are the position and speed of the puck at t = 7.00 s?

A 4.50-kg experimental cart undergoes an acceleration in a straight line (the x-axis). The graph in Fig. E4.13 shows this acceleration as a function of time. 

(a) Find the maximum net force on this cart. When does this maximum force occur? (b) During what times is the net force on the cart a constant? (c) When is the net force equal to zero?

In a head-on auto collision, passengers who are not wearing seat belts may be thrown through the windshield. Use Newton’s laws of motion to explain why this happens.

In a head-on collision between a compact 1000-kg car and a large 2500-kg car, which one experiences the greater force? Explain. Which one experiences the greater acceleration? Explain why. Why are passengers in the small car more likely to be injured than those in the large car, even when the two car bodies are equally strong?

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

or the sprinter herself?

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  $12{\text{\hspace{0.17em}m/s}}^{\text{2}}$?

(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 mg. How many newtons does it weigh? 

(c) A house cat typically weighs 45 N. How many pounds does it weigh, and what

is its mass in kilograms?

At the surface of Jupiter’s moon Io, the acceleration due to gravity is  $g=1.81{\text{\hspace{0.17em}m/s}}^{\text{2}}$. A watermelon weighs 44.0 N at the surface of the earth. 

(a) What is the watermelon’s mass on the earth’s surface? 

(b) What would be its mass and weight on the surface of Io?

An electron (mass = $\mathbf{9}\mathbf{.}\mathbf{11}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{31}}\mathbf{kg}$ ) 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 cm away. It reaches the grid with a speed of . If the accelerating force is constant, compute 

(a) the acceleration; 

(b) the time to reach the grid; and 

(c) the net force, in newtons. Ignore the gravitational force on the electron.

A 2.75-kg cat moves in a straight line (the x-axis). Figure E4.14 shows a graph of the x-component of this cat’s velocity as a function of time. (a) Find the maximum net force on this cat. When does this force occur? (b) When is the net force on the cat equal to zero? (c) What is the net force at time 8.5 s?

A small 8.00-kg rocket burns fuel that exerts a time-varying upward force on the rocket (assume constant mass) as the rocket moves upward from the launch pad. This force obeys the equation $\mathbf{F}\mathbf{=}\mathbf{A}\mathbf{+}{\mathbf{Bt}}^{\mathbf{2}}$  . Measurements show that at t = 0, the force is 100.0 N, and at the end of the first 2.00 s, it is 150.0 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 the fuel ignites. (c) Suppose that you were using this rocket in outer space, far from all gravity. What would its acceleration be 3.00 s after fuel ignition?

An astronaut’s pack weighs 17.5 N when she is on the earth but only 3.24 N when she is at the surface of a moon.

(a) What is the acceleration due to gravity on this moon?

(b) What is the mass of the pack on this moon?

Question: The upward normal force exerted by the floor is 620 N on an elevator passenger who weighs 650 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?

Boxes A and B are in contact on a horizontal, frictionless surface (Fig. E4.23). Box A has mass $\mathbf{20}\mathbf{ }\mathbf{kg}$ and box B has mass$\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{kg}$. A horizontal force of $\mathbf{250}\mathbf{N}$ is exerted on box A. What is the magnitude of the force that box A exerts on box B?

 A student of mass $\mathbf{45}\mathbf{ }\mathbf{kg}$ jumps off a high diving board. What is the acceleration of the earth toward her as she accelerates towards the earth with an acceleration of  $\mathbf{9}\mathbf{.}\mathbf{8}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$? Use  $\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{24}}$ for the mass of the earth, and assume that the net force on the earth is the force of gravity she exerts on it.

Crates A and B sit at rest side by side on a frictionless horizontal surface. They have masses ${\mathbf{m}}_{\mathbf{A}}$ and ${\mathbf{m}}_{\mathbf{B}}$ respectively. When a horizontal force  $\stackrel{\rightharpoonup }{\mathbf{F}}$ is applied to crate A, 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 $\overrightarrow{\mathbf{F}}$ is less than the total weight of the two crates, will it cause the crates to move? Explain.

You pull horizontally on block B in Fig. E4.26, causing both blocks to move together as a unit. For this moving system, make a carefully labeled free-body diagram of block A if (a) the table is frictionless and (b) there is friction between block B and the table and the pull is equal in magnitude to the friction force on block B due to the table.  

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 chair of mass $\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{kg}$ is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force $\mathbf{F}\mathbf{=}\mathbf{40}\mathbf{.}\mathbf{0}\mathbf{\hspace{0.33em}}\mathbf{N}$ that is directed at an angle of $\mathbf{37}\mathbf{.}\mathbf{0}\mathbf{°}$ 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 large box containing your new computer sits on the bed of your pickup truck. You are stopped at a red light. When the light turns green, you stomp on the gas and the truck accelerates. To your horror, the box starts to slide towards 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 horizontal truck bed is not frictionless.)

A 5.60 kg bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 75.0 N. If the bucket starts from rest, what is the minimum time required to raise the bucket a vertical distance of 12.0 m without breaking the cord?

 You have just landed on Planet X. You release a $\mathbf{100}\mathbf{ }\mathbf{g}$ ball from rest from a height of $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$ and measures that it takes $\mathbf{3}\mathbf{.}\mathbf{40}\mathbf{ }\mathbf{s}$ to reach the ground. Ignore any force on the ball from the atmosphere of the planet. How much does the $\mathbf{100}\mathbf{ }\mathbf{g}$ ball weigh on the surface of Planet X?

Two adults and a child want to push a wheeled cart in the direction marked x in Fig. P4.33. The two adults push with horizontal forces ${\overrightarrow{\mathbf{F}}}_{\mathbf{1}}$ and ${\overrightarrow{\mathbf{F}}}_{\mathbf{2}}$ as shown. (a) Find the magnitude and direction of the smallest force that the child should exert. Ignore the effects of friction. (b) If the child exerts the minimum force found in part (a), the cart accelerates at $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$ in the +x direction. What is the weight of the cart?

Basketball player Darrell Griffith is on record as attaining a standing vertical jump of 1.2 m(4 ft). (This means that he moved upward by 1.2m after his feet left the floor.) Griffith weighed 890 N(200lb). (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 was his average acceleration (magnitude and direction) while he pushed against the floor? (c) Draw his free-body diagram. In terms of the forces on the diagram, what was the net force on him? Use Newton’s laws and the results of part (b) to calculate the average force he applied to the ground.

 An advertisement claims that a particular automobile can “stop on a dime.” What net force would be necessary to stop $\mathbf{850}\mathbf{-}\mathbf{kg}$ automobile traveling initially at $\mathbf{45}\mathbf{.}\mathbf{0}\mathbf{km}\mathbf{/}\mathbf{h}$ in a distance equal to the diameter of a dime, $\mathbf{1}\mathbf{.}\mathbf{8}\mathbf{cm}$?

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