Dynamics: Force and Newton's Laws of Motion

Near the end of a marathon race, the first two runners are separated by a distance of 45 m. The front runner has a velocity of 3.50 m/s, and the second a velocity of 4.20 m/s. (a) what is the velocity of the second runner relative to the first? (b) If the front runner is 250 m from the finish line, who will win the race, assuming they run at constant velocity? (c) What distance ahead will the winner be when she crosses the finish line?

Suppose two children push horizontally, but in exactly opposite directions, on a third child in a wagon. The first child exerts a force of 75.0 N, the second a force of 90.0 N, friction is 12.0 N, and the mass of the third child plus wagon is 23.0 kg. 

(a) What is the system of interest if the acceleration of the child in the wagon is to be calculated? 

(b) Draw a free-body diagram, including all forces acting on the system. 

(c) Calculate the acceleration. 

(d) What would the acceleration be if friction were 15.0 N?

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.

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

How are inertia and mass related?

 What is the relationship between weight and mass? Which is an intrinsic, unchanging property of a body?

 Why can we neglect forces such as those holding a body together when we apply Newton’s second law of motion?

Describe a situation in which the net external force on a system is not zero, yet its speed remains constant.

A system can have a nonzero velocity while the net external force on it is zero. Describe such a situation.

A rock is thrown straight up. What is the net external force acting on the rock when it is at the top of its trajectory?

(a) Give an example of different net external forces acting on the same system to produce different accelerations. (b) Give an example of the same net external force acting on systems of different masses, producing different accelerations. (c) What law accurately describes both effects? State it in words and as an equation.

If the acceleration of a system is zero, are no external forces acting on it? What about internal forces? Explain your answers.

If a constant, nonzero force is applied to an object, what can you say about the velocity and acceleration of the object?

The gravitational force on the basketball in Figure 4.6 is ignored. When gravity is taken into account, what is the direction of the net external force on the basketball—above horizontal, below horizontal, or still horizontal.

When you take off in a jet aircraft, there is a sensation of being pushed back into the seat. Explain why you move backward in the seat—is there really a force backward on you? (The same reasoning explains whiplash injuries, in which the head is apparently thrown backward.)

A device used since the 1940 s to measure the kick or recoil of the body due to heart beats is the “ballistocardiograph.” What physics principle(s) are involved here to measure the force of cardiac contraction? How might we construct such a device?

Describe a situation in which one system exerts a force on another and, as a consequence, experiences a force that is equal in magnitude and opposite in direction. Which of Newton’s laws of motion apply?

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

Newton’s third law of motion tells us that forces always occur in pairs of equal and opposite magnitude. Explain how the choice of the “system of interest” affects whether one such pair of forces cancels.

 If a leg is suspended by a traction setup as shown in Figure 4.30, what is the tension in the rope?

 In a traction setup for a broken bone, with pulleys and rope available, how might we be able to increase the force along the femur using the same weight? (See Figure 4.30.) (Note that the femur is the shin bone shown in this image.

To simulate the apparent weightlessness of space orbit, astronauts are trained in the hold of a cargo aircraft that is accelerating downward at g. Why will they appear to be weightless, as measured by standing on a bathroom scale, in this accelerated frame of reference? Is there any difference between their apparent weightlessness in orbit and in the aircraft?

A cartoon shows the toupee coming off the head of an elevator passenger when the elevator rapidly stops during an upward ride. Can this really happen without the person being tied to the floor of the elevator? Explain your answer.

 Explain, in terms of the properties of the four basic forces, why people notice the gravitational force acting on their bodies if it is such a comparatively weak force.

What is the dominant force between astronomical objects? Why are the other three basic forces less significant over these very large distances?

Give a detailed example of how the exchange of a particle can result in an attractive force. (For example, consider one child pulling a toy out of the hands of another.)

A 63.0 kg sprinter starts a race with an acceleration of 4.20 m/s². What is the net external force on him?

Since astronauts in orbit are apparently weightless, a clever method of measuring their masses is needed to monitor their mass gains or losses to adjust diets. 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 the astronaut’s acceleration is measured to be 0.893 m/s².

(a) Calculate her mass.

(b) By exerting a force on the astronaut, the vehicle in which they orbit experiences an equal and opposite force. Discuss how this would affect the measurement of the astronaut’s acceleration. Propose a method in which recoil of the vehicle is avoided.

A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate the magnitude of its acceleration.

In Figure 4.7, the net external force on the 24-kg mower is stated to be 51 N. If the force of friction opposing the motion is 24 N, what force F (in newtons) is the person exerting on the mower? Suppose the mower is moving at 1.5 m/s when the force F is removed. How far will the mower go before stopping?

(a) If the rocket sled shown in Figure 4.32 starts with only one rocket burning, what is the magnitude of its acceleration? Assume that the mass of the system is 2100 kg, the thrust T is 2.4×10⁴ N, and the force of friction opposing the motion is known to be 650 N. 

(b) Why is the acceleration not one-fourth of what it is with all rockets burning?

The rocket sled shown in Figure 4.33 accelerates at a rate of 49.0 m/s². 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 by using a ratio. 

(b) Calculate the direction and magnitude of the total force the seat exerts against his body.

Suppose the mass of a fully loaded module in which astronauts take off from the Moon is 10,000 kg. The thrust of its engines is 30,000 N. 

(a) Calculate its the magnitude of acceleration in a vertical takeoff from the Moon. 

(b) Could it lift off from Earth? If not, why not? If it could, calculate the magnitude of its acceleration.

A brave but inadequate rugby player is being pushed backward by an opposing player who is exerting a force of 800 N on him. The mass of the losing player plus equipment is 90.0 kg, and he is accelerating at 1.20 m/s² backward. (a) What is the force of friction between the losing player’s feet and the grass? (b) What force does the winning player exert on the ground to move forward if his mass plus equipment is 110 kg? 

(c) Draw a sketch of the situation showing the system of interest used to solve each part. For this situation, draw a free-body diagram and write the net force equation.

Two teams of nine members each engage in a tug of war. Each of the first team’s members has an average mass of 68 kg and exerts an average force of 1350 N horizontally. Each of the second team’s members has an average mass of 73 kg and exerts an average force of 1365 N horizontally. 

(a) What is magnitude of the acceleration of the two teams? 

(b) What is the tension in the section of rope between the teams?

A freight train consists of two 8.00×10⁴ -kg engines and 45 cars with average masses of 5.50×10⁴ kg.

(a) What force must each engine exert backward on the track to accelerate the train at a rate of 5.00×10–2 m/s² if the force of friction is 7.50×10⁵ N, assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently trains are very energy-efficient transportation systems.

(b) What is the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?

Commercial airplanes are sometimes pushed out of the passenger loading area by a tractor. 

(a) An 1800-kg tractor exerts a force of 1.75×10⁴ N backward on the pavement, and the system experiences forces resisting motion that total 2400 N. If the acceleration is 0.150 m/s², what is the mass of the airplane?

(b) Calculate the force exerted by the tractor on the airplane, assuming 2200 N of the friction is experienced by the airplane. 

(c) Draw two sketches showing the systems of interest used to solve each part, including the free-body diagrams for each.

A 1100-kg car pulls a boat on a trailer.

(a) What total force resists the motion of the car, boat, and trailer, if the car exerts a 1900-N force on the road and produces an acceleration of 0.550 m/s²? The mass of the boat plus trailer is 700 kg. 

(b) What is the force in the hitch between the car and the trailer if 80% of the resisting forces are experienced by the boat and trailer?

(a) Find the magnitudes of the forces F₁ and F₂ that add to give the total force F_tot shown in Figure 4.35. This may be done either graphically or by using trigonometry. 

(b) Show graphically that the same total force is obtained independent of the order of addition of F₁ and F₂ . 

(c) Find the direction and magnitude of some other pair of vectors that add to give F_tot . Draw these to scale on the same drawing used in part (b) or a similar picture.

An American football lineman reasons that it is senseless to try to out-push the opposing player, since no matter how hard he pushes he will experience an equal and opposite force from the other player. Use Newton’s laws and draw a free-body diagram of an appropriate system to explain how he can still out-push the opposition if he is strong enough.

(a) Calculate the tension in a vertical strand of spider web if a spider of mass 8.00×10⁻⁵ 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 4.17. The strand sags at an angle of 12º below the horizontal. Compare this with the tension in the vertical strand (find their ratio).

Suppose a 60.0-kg gymnast climbs a rope. 

(a) What is the tension in the rope if he climbs at a constant speed? 

(b) What is the tension in the rope if he accelerates upward at a rate of 1.50 m/s²?

Consider the baby being weighed in Figure 4.34. 

(a) What is the mass of the child and basket if a scale reading of 55 N is observed? 

(b) What is the tension T1 in the cord attaching the baby to the scale? 

(c) What is the tension T2 in the cord attaching the scale to the ceiling, if the scale has a mass of 0.500 kg? 

(d) Draw a sketch of the situation indicating the system of interest used to solve each part. The masses of the cords are negligible.

Suppose your car was mired deeply in the mud, and you wanted to use the method illustrated in Figure 4.37 to pull it out. 

(a) What force would you have to exert perpendicular to the center of the rope to produce a force of 12,000 N on the car if the angle is 2.00°? In this part, explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. 

(b) Real ropes stretch under such forces. What force would be exerted on the car if the angle increases to 7.00° and you still apply the force found in part (a) to its center?

Figure 4.39 shows Superhero and Trusty Sidekick hanging motionless from a rope. Superhero’s mass is 90.0 kg, while Trusty Sidekick’s is 55.0 kg, and the mass of the rope is negligible. 

(a) Draw a free-body diagram of the situation showing all forces acting on Superhero, Trusty Sidekick, and the rope. 

(b) Find the tension in the rope above Superhero. 

(c) Find the tension in the rope between Superhero and Trusty Sidekick. Indicate on your free-body diagram the system of interest used to solve each part.

A nurse pushes a cart by exerting a force on the handle at a downward angle 35.0º below the horizontal. The loaded cart has a mass of 28.0 kg, and the force of friction is 60.0 N.

(a) Draw a free-body diagram for the system of interest. 

(b) What force must the nurse exert to move at a constant velocity?

Unreasonable Results 

(a) Repeat Exercise 4.29, but assume an acceleration of 1.20 m/s2 is produced. (b) What is unreasonable about the result? 

(c) Which premise is unreasonable, and why is it unreasonable?

Unreasonable Results 

(a) What is the initial acceleration of a rocket that has a mass of 1.50×10⁶ kg at takeoff, the engines of which produce a thrust of 2.00×10⁶ N ? Do not neglect gravity. 

(b) What is unreasonable about the result? (This result has been unintentionally achieved by several real rockets.)

(c) Which premise is unreasonable, or which premises are inconsistent? (You may find it useful to compare this problem to the rocket problem earlier in this section.)

Consider the tension in an elevator cable during the time the elevator starts from rest and accelerates its load upward to some cruising velocity. Taking the elevator and its load to be the system of interest, draw a free-body diagram. Then calculate the tension in the cable. Among the things to consider are the mass of the elevator and its load, the final velocity, and the time taken to reach that velocity.

Consider two people pushing a toboggan with four children on it up a snow-covered slope. Construct a problem in which you calculate the acceleration of the toboggan and its load. Include a free-body diagram of the appropriate system of interest as the basis for your analysis. Show vector forces and their components and explain the choice of coordinates. Among the things to be considered are the forces exerted by those pushing, the angle of the slope, and the masses of the toboggan and children.