Chapter 4: Dynamics: Force and Newton's Laws of Motion

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.

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.

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?

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.

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 F₁and 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.

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.