Problem-Solving Strategies

Applying Newton’s Second Law

$F_{\text{net}x}=\text{ma},$

$F_{\text{net}y}=0.$

A $5\text{.}\text{00}\times {\text{10}}^5\text{-kg}$

rocket is accelerating straight up. Its engines produce $1\text{.}\text{250}\times {\text{10}}^7\text{N}$

of thrust, and air resistance is $4\text{.}\text{50}\times {\text{10}}^6\text{N}$

. What is the rocket’s acceleration? Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion.

Using the free-body diagram:

$F_{\text{net}}=T-f-mg=\text{ma}$

,

so that

$a=\frac{T-f-\text{mg}}{m}=\frac{1\text{.}\text{250}\times {\text{10}}^7\text{N}-4.50\times {\text{10}}^{\text{6}}N-(5.00\times {\text{10}}^5\text{kg})(9.{\text{80 m/s}}^2)}{5.00\times {\text{10}}^5\text{kg}}=\text{6.20}{\text{m/s}}^2$

.

The wheels of a midsize car exert a force of 2100 N backward on the road to accelerate the car in the forward direction. If the force of friction including air resistance is 250 N and the acceleration of the car is $1\text{.}{\text{80 m/s}}^2$

, what is the mass of the car plus its occupants? Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. For this situation, draw a free-body diagram and write the net force equation.

Calculate the force a 70.0-kg high jumper must exert on the ground to produce an upward acceleration 4.00 times the acceleration due to gravity. Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion.

1. Use Newton’s laws of motion.

2. Given : $a=4.00g=(4.00)(9.{\text{80 m/s}}^2)=\text{39.2}{\text{m/s}}^2\text{;}$ $m=\text{70}\text{.}\text{0 kg}$

   , Find: $F$

   .

3. $$ $$ $\sum F\text{=+}F-w=\text{ma}\text{,}$

   so that

   $F=\text{ma}+w=\text{ma}+\text{mg}=m(a+g)$

   .

   $F=(\text{70.0 kg})[(\text{39}\text{.}{\text{2 m/s}}^2)+(9\text{.}{\text{80 m/s}}^2)]$ $=3.\text{43}\times {\text{10}}^3\text{N}$

   . The force exerted by the high-jumper is actually down on the ground, but $F$

   is up from the ground and makes him jump.

4. This result is reasonable, since it is quite possible for a person to exert a force of the magnitude of ${\text{10}}^3\text{N}$

   .

When landing after a spectacular somersault, a 40.0-kg gymnast decelerates by pushing straight down on the mat. Calculate the force she must exert if her deceleration is 7.00 times the acceleration due to gravity. Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion.

A freight train consists of two $8.00\times {10}^4\text{-kg}$

engines and 45 cars with average masses of $5.50\times {10}^4\text{kg}$

. (a) What force must each engine exert backward on the track to accelerate the train at a rate of $5.00\times {\text{10}}^{\text{–2}}{\text{m/s}}^2$

if the force of friction is $7\text{.}\text{50}\times {\text{10}}^5\text{N}$

, 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?

(a) $4\text{.}\text{41}\times {\text{10}}^5\text{N}$

(b) $1\text{.}\text{50}\times {\text{10}}^5\text{N}$

Commercial airplanes are sometimes pushed out of the passenger loading area by a tractor. (a) An 1800-kg tractor exerts a force of $1\text{.}\text{75}\times {\text{10}}^4\text{N}$

backward on the pavement, and the system experiences forces resisting motion that total 2400 N. If the acceleration is $0\text{.}{\text{150 m/s}}^2$

, 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 $0\text{.}{\text{550 m/s}}^2$

? 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) $\text{910 N}$

(b) $1\text{.}\text{11}\times {\text{10}}^3\text{N}$

(a) Find the magnitudes of the forces ${\mathbf{\text{F}}}_1$

and ${\mathbf{\text{F}}}_2$

that add to give the total force ${\mathbf{\text{F}}}_{\text{tot}}$

shown in [link]. 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 ${\mathbf{\text{F}}}_1$

and ${\mathbf{\text{F}}}_2$

. (c) Find the direction and magnitude of some other pair of vectors that add to give ${\mathbf{\text{F}}}_{\text{tot}}$

. Draw these to scale on the same drawing used in part (b) or a similar picture.

Two children pull a third child on a snow saucer sled exerting forces ${\mathbf{\text{F}}}_1$

and ${\mathbf{\text{F}}}_2$

as shown from above in [link]. Find the acceleration of the 49.00-kg sled and child system. Note that the direction of the frictional force is unspecified; it will be in the opposite direction of the sum of ${\mathbf{\text{F}}}_1$

and ${\mathbf{\text{F}}}_2$

.

$a=\text{0.139 m/s}$

, $\theta =\mathrm{12.4º}$

north of east

Suppose your car was mired deeply in the mud and you wanted to use the method illustrated in [link] 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?

What force is exerted on the tooth in [link] if the tension in the wire is 25.0 N? Note that the force applied to the tooth is smaller than the tension in the wire, but this is necessitated by practical considerations of how force can be applied in the mouth. Explicitly show how you follow steps in the Problem-Solving Strategy for Newton’s laws of motion.

1. Use Newton’s laws since we are looking for forces.

2. Draw a free-body diagram:

3. The tension is given as $T=\text{25.0 N.}$

   Find

   $F_{\text{app}}\text{.}$

   Using Newton’s laws gives:

   ${\text{Σ F}}_y=\mathrm{0,}$

   so that applied force is due to the y-components of the two tensions:

   $F_{\text{app}}=2T\text{sin}\text{θ}=2(\text{25.0 N})\text{sin}\left(\text{15º}\right)=\text{12.9 N}$

   The x-components of the tension cancel. $\sum F_x=0$

   .

4. This seems reasonable, since the applied tensions should be greater than the force applied to the tooth.

[link] 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 $\text{35.0º}$

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?

Construct Your Own Problem 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.

Construct Your Own Problem 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.

Unreasonable Results (a) Repeat [link], but assume an acceleration of $1\text{.}{\text{20 m/s}}^2$

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 $1\text{.}\text{50}\times {\text{10}}^6\text{kg}$

at takeoff, the engines of which produce a thrust of $2\text{.}\text{00}\times {\text{10}}^6\text{N}$

? 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.)