Newtons Law of Motion

Chapter 4 Newton's Laws of Motion

Fundamentals of Physics: A Senior Course

SHORT ANSWER

1.An object with a mass of 15 kg rests on a frictionless horizontal plane and is acted upon by a horizontal force of 30 N.

(a) What is its acceleration?

(b) How far will it move in 10 s?

(c) What will be its velocity after 10 s?

ANS:

(a) Using Newtons Second Law,

(b) Since the acceleration is constant, the equations of uniformly accelerated motion may be used to describe its motion.

(c)

REF:K/UOBJ:2.2LOC:FMV.01KEY:FOP 4.4, p.136

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2.(a) What is the net force required to give an automobile of mass 1600 kg an acceleration of 4.5 m/s2?

(b) What is the acceleration of a wagon of mass 20 kg if a horizontal force of 64 N is applied to it (ignore friction)?

(c) What is the mass of a block of iron if a net force of 240 N causes it to accelerate across a smooth horizontal surface at 2.5 m/s2?

ANS:

(a)

(b)

(c)


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3.A 1.0 kg toy car is moving across a smooth floor with a velocity of 5.0 m/s. An unbalanced force of 2.0 N acts on the car for 4.0 s. Determine the velocity of the car at the end of the interval in each of the following cases.

(a) if the force acts in the direction of motion of the car

(b) if the force acts in the opposite direction to the motion of the car

ANS:

(a)

(b)


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4.State the reaction force for each of the following forces.

(a) the southward force of a field goal kickers toe on a football

(b) the backward force of a joggers shoe on the ground

(c) the downward force of a book on a desk

(d) the backward force of a jets engines on its exhaust gases

(e) the backward pull of a swimmers hands on the water in the butterfly stroke

ANS:

(a) the northward force of the football on the kickers toe

(b) the forward force of the ground on the joggers shoe

(c) the upward force of the desk on the book

(d) the forward force of the exhaust gases on a jet engine

(e) the forward force of the water on the swimmers hands

REF:C, K/UOBJ:2.2LOC:FMV.01KEY:FOP 4.6, p.144

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5.A beginning physics student, confused by a seeming contradiction in Newtons laws, asks her teacher the following question: If, for every force there is an equal and opposite reaction force, then all forces in nature come in equal and opposite pairs, and are therefore balanced. Thus, since there can never be such a thing as an unbalanced force, how can any object ever accelerate? Explain the fault in this common misconception.

ANS:

For forces to be balanced, they must act on the same object and be equal and opposite. The reaction to any force does not act on the same object as the force; it acts in the opposite direction on the agent exerting the original force. Thus, when considering the motion of one object, we look at the force, and when considering the motion of the agent of this force, we look at the reaction.


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6.A fireman at the scene of a fire is holding a heavy hose out of which water is gushing. To keep his balance, he often has to lean. Which way does he lean, forward or backward, and why?

ANS:

The fireman has to lean forward. The hose pushes the water forward (the force) so the water pushes backward on the hose with an equal force (the reaction). To keep from being pushed over backward by the reaction force, the fireman leans forward.

REF:C, MCOBJ:2.2LOC:FMV.01KEY:FOP 4.6, p.145

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7.A squirrel with an armful of nuts is sliding helplessly across the flat, icy roof, getting dangerously close to the edge. He understands Newtons Third Law, and is able to save himself. Explain how he does it.

ANS:

He saves himself by throwing nuts towards the edge of the roof. To do so, he exerts a force on each nut towards the edge. The nut exerts an equal force back on the squirrel, away from the edge, causing him to slow down.


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8.A rocket carrying the Space Shuttle blasts off from the Kennedy Space Center. In which of the following positions will it have a greater acceleration, assuming that the engines exert a constant force?

(a) at ground level

(b) 50 m above ground level

Explain your answer.

ANS:

If the engine exerts a constant force down on the exhaust gases, the exhaust gases exert a constant force up on the engines and, hence, the entire rocket. However, as fuel is burned, the rockets mass decreases, and since , the rocket will have a greater acceleration 50 m above the ground than at ground level.

REF:C, MCOBJ:2.2LOC:FMV.03KEY:FOP 4.6, p.145

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9.Two crates, of mass 12.0 kg and 20.0 kg, respectively, are pushed across a smooth floor together, the 20 kg crate in front of the 12 kg crate. Their acceleration is 1.75 m/s2. Calculate each of the following.

(a) the force applied to push the crates

(b) the action-reaction forces between the two crates

Recalculate (a) and (b) if the relative positions of the two crates are reversed.

ANS:

(a)

(b) Consider the 20 kg crate only. The force accelerating it is the force exerted on it by the 12.0 kg crate, .

For the reversed crates, is unchanged but


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10.A locomotive with a mass of 2.0 104 kg accelerates from rest to a velocity of 2.0 m/s in 5.0s. If it is pulling a train of 20 cars, each of mass 1.0 104 kg, what is the force in the coupling at each of the following points?

(a) between the locomotive and the first car

(b) between the 10th car and the 11th car

ANS:

(a) For the entire train,

For the force between the locomotive and the 1st car, the total mass being accelerated by this force is 20 cars of mass 1.0 kg each.

(b) For the force between the 10th and 11th car,

REF:K/U, MCOBJ:2.3LOC:FMV.01KEY:FOP 4.6, p.147

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11.Three small children of mass 20.0 kg, 24.0 kg, and 16.0 kg, respectively, hold hands, as shown, and are pulled across a smooth frozen pond by a larger boy on skates, who pulls a horizontal rope being held by the first child. The skater pulls on the rope with a force of 135 N. Calculate each of the following.

(a) the acceleration of the skater

(b) the force with which each pair of children must hold hands, to ensure that the chain is not broken

ANS:

For the acceleration of the entire group,

(b) For the force between the 20.0 kg and the 24.0 kg child, mass being accelerated is 16.0 kg + 24.0 kg.

Similarly, for the force between the 24.0 kg and 16.0 kg child,

Free body diagrams of the 3 children appear as:


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12.A 3.0 kg toy is pulled by a force of 24 N. If the toy starts from rest, how far will it travel in the first 5.0 s?

ANS:


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13.A 40 kg sprinter starts from rest and 2.0 s later is running at a speed of 8.0 m/s. What is the average net horizontal force acting on her? What exerts this force?

ANS:

This forward force is exerted on the sprinter by the ground and is the reaction to the backward force that her feet exert on Earth.

REF:K/U, COBJ:2.2LOC:FMV.01KEY:FOP 4.8, p.151

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14.An 8.0 g bullet travelling at 400 m/s passes through a heavy block of wood in 4.0 104 s, emerging with a velocity of 100 m/s. Ignore any motion of the wood.

(a) With what average force did the wood oppose the motion of the bullet?

(b) How thick is the block of wood?

ANS:

(a)

(b)


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15.A 0.22 calibre rifle shoots a bullet of mass 1.8 g with a muzzle velocity of 500 m/s. If the barrel is 25 cm long, what is the average force exerted on the bullet while it is in the barrel?

ANS:


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16.A sled of 6.0 kg mass is moving along a smooth, horizontal ice surface with a velocity of v0. A force of 36 N is applied to the sled in its direction of motion, increasing its velocity to 2v0 while it moves 10 m. Find the following.

(a) the sleds original velocity, v0

(b) the length of time that the force acted.

ANS:

(a)

(b)


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17.A gardener pushes down along the handle of a lawn mower of 20 kg mass with a force of 150 N. The handle makes an angle of 60 with the ground. Calculate the instantaneous acceleration of the mower if the frictional force between its wheels and the ground at that instant is 25 N.

ANS:

Breaking the applied 150 N force into horizontal and vertical components:

The net horizontal force accelerating the mower is:


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18.A man drags a package across the floor with a force of a 40 N, as shown. The mass of the package is 10 kg. If the acceleration of the package is 3.5 m/s2, and friction can be neglected, at what angle to the horizontal does the man pull?

ANS:

If the angle of the between the rope and horizontal is q, the net horizontal force on the package is:

But


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19.A boy with a mass of 30 kg pulls a cart with a mass of 100 kg towards himself by a rope. With what force does he have to pull on the rope to accelerate the cart at 2.0 m/s2? With what force must his feet push on the ground to keep him from moving towards the cart? If there is no friction between his feet and the ground, what is his acceleration?

ANS:

For the cart,

For no acceleration of the boy, Fnet = 0.

Since the rope exerts a force of 2.0 102 N on the boy, his feet must have an opposing force of 2.0102N.


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20.A cart is pulled in each of several trials, with a different number of stretched elastic bands. A constant acceleration is observed in each trial. The graph shows acceleration versus force exerted by the stretched bands.

(a) What is the mass of the cart?

(b) An extrapolation of the graph does not pass through the origin. What does this indicate?

ANS:

(a)

(b) The force on the cart when a = 0 is 0.60 N. This represents the constant force of friction on the cart.

REF:K/U, IOBJ:2.2LOC:FM2.01KEY:FOP 4.8, p.153

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21.A car with a mass of 1000 kg is moving in a straight line at a constant speed of 30 m/s. It is brought to rest in 25 s. What constant force is acting to stop the car?

ANS:


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22.An electron has a mass of 9.1 1031 kg. Between the electrodes of a cathode-ray tube, it moves a distance of 4.0 mm, accelerated by a net electrical force of 5.6 1015 N. Assuming that it started from rest, find its acceleration and its final velocity.

ANS:


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23.A speedboat is pulling two water skiers, using two ropes attached to the back of the boat. Each rope makes an angle of 30 on either side of the boats axis. The force exerted by each rope is 400 N. If the boat is moving in a straight line at constant speed, what force must it be exerting to keep the pair of skiers moving?

ANS:

If the boat pulls on each skier with 400 N, each skier pulls back with 400 N on the boat, as shown. Also, if the boat pushes back on the water with , the water pushes forward with on the boat, as shown. Since the boat is moving in a straight line, at a steady speed,

Taking the components in the direction of the boat,


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24.A block of mass 2.0 kg is placed on a smooth plane, inclined to the horizontal at an angle of 15. The force of gravity acting straight down on the block is 20 N.

(a) What is the acceleration of the block down the plane?

(b) How far up the plane was the block released, if it took 1.5 s to reach the bottom after it was released from rest?

ANS:

(a)

The force of gravity has two components: to the plane, and parallel to the plane. The plane exerts a normal force equal and opposite to . accelerated the block down the plane.

For the magnitude of

(b)


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PROBLEM

1.A 0.50 kg skateboard is at rest on a rough, level floor on which two lines have been drawn 1.0 m apart. A constant horizontal force is applied to the skateboard at the beginning of the interval, and is removed at the end. The skateboard takes 8.5 s to travel the 1.0 m distance, and it then coasts for another 1.25 m before coming to rest. Calculate the force applied to the skateboard, and also the constant frictional force opposing its motion.

ANS:

For the first interval of motion, when Fa and Ff are acting:

For the second interval of motion, when only Ff is acting:

REF:K/U, IOBJ:2.4LOC:FMV.01KEY:FOP 4.4, p.138

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2.A small boy pulls his wagon, of mass 24 kg, giving it a horizontal acceleration of 1.5 m/s2. If the wagons handle makes an angle of 40 with the ground while the boy is pulling on it, and there is a frictional force of 6.0 N opposing the wagons motion, with what force is he pulling on the handle of the wagon?

ANS:


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3.Forces of 100 N [N] and 80 N [W] act simultaneously on an object of mass 10 kg. What is the acceleration of the object?

ANS:

Using the vector diagram shown,


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4.Forces of 2.0 N and 1.0 N act on an object of mass 5.0 kg, as shown in the diagram.

(a) Calculate the net force acting on the object.

(b) What is the acceleration of the object?

ANS:

(a) Using the vector diagram shown:

Using the cosine law:

Using the sine law:

(b)


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5.An empty railway flatcar of mass 15 000 kg is being pulled along a smooth, horizontal track by a tractor travelling on a road parallel to the track. The rope joining the tractor and the flatcar makes an angle of 25 with the track.

(a) If the acceleration of the flatcar is 0.80 m/s2, what is the force exerted by the rope on the flatcar?

(b) Why does the flatcar have no sideways acceleration? A sideways force is exerted by the rope!

ANS:

(a) For the flatcar,

(b) The track exerts an equal force in the opposite direction to the sideways force of the rope (actually 5.1 103 N).


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6.Two girls, one of mass 40 kg and the other of mass 60 kg, are standing side by side in the middle of a frozen pond. One pushes the other with a force of 360 N for 0.10 s. The ice is essentially frictionless.

(a) What is each girls acceleration?

(b) What velocity will each girl acquire in the 0.10 s that the force is acting?

(c) How far will each girl move during the same time period?

ANS:

(a)

(b)

(c)


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7.A motorist has a reaction time of 0.60 s. (Reaction time is the interval between seeing a danger and applying the brakes.) While driving at 72 km/h, he sees a child run suddenly onto the road, 40 m in front of his car. If the mass of the car is 1000 kg and the average horizontal force supplied during braking is 8000 N, will he be able to stop in time to avoid hitting the child?

ANS:

The car stopped 3.0 m short of hitting the child.


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8.A childs wagon experiences a frictional force of 73 N whenever it is in motion, regardless of the load it is carrying. An applied horizontal force of 128 N causes the wagon to accelerate at 5.0 m/s2. The same applied force, with a child on the wagon, causes it to accelerate at 1.0 m/s2. What is the mass of the child?

ANS:

Without the child,

With the child,

The mass of the child is 55 kg 11 kg = 44 kg.


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9.A net force of 8.0 N gives a mass m1 and acceleration of 2.0 m/s2 and a mass m2, an acceleration of 4.0 m/s2. What acceleration would the force give the two masses if they were fastened together?

ANS:


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10.The graph shows the velocity of a 5.0 kg radio-controlled toy car, moving in a straight line, as a function of time. Plot a force-time graph for the car.

ANS:

For any interval,


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11.Two girls pull a sled across a field of snow, as shown in the diagram. A third girl pulls backward with a 2.0 N force. If the mass of the sled is 10 kg, determine its instantaneous acceleration.

ANS:

The net force acting on the sled is the vector sun of the three forces acting:

Using components in the x-y direction:


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12.A plane takes off from a level runway with two gliders in tow, one behind the other. The first glider has a mass of 1600 kg and the second a mass of 800 kg. The frictional drag may be assumed as constant and equal to 2000 N on each glider. The towrope between the first glider and the plane can withstand a tension of 10 000 N.

(a) If a velocity of 40 m/s is required for takeoff, how long a runway is needed?

(b) How strong must the towrope between the two gliders be?

ANS:

(a)

Then, if

(b) For the second glider,


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13.A baby carriage with a mass of 50 kg is being pushed along a rough sidewalk with an applied horizontal force of 200 N, and it has a constant velocity of 3.0 m/s.

(a) What other horizontal force is acting on the carriage, and what is the magnitude of that force?

(b) What value of applied horizontal force would be required to accelerate the carriage from rest to 7.0 m/s in 2.0 s?

ANS:

(a) The baby carriage is being pushed along a rough sidewalk; therefore, there must be some frictional force that is opposing its motion, between its wheels and the concrete. However, since the carriage is moving with a constant velocity (zero acceleration), according to Newtons Second Law the net force acting on it must be zero. Thus, the other force acting on the carriage is the force of friction, and its value is 200 N in the direction opposite to the carriages motion.

(b) The acceleration of the carriage is given by

The net horizontal force in the direction of this acceleration is


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14.A bullet of mass 20 g strikes a fixed block of wood at a speed of 320 m/s. The bullet embeds itself in the block of wood, penetrating to a depth of 6.0 cm. Calculate the average net force acting on the bullet while it is being brought to rest.

ANS:


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Newtons Law of Motion