11.14 Summary and exercises  (Page 15/2)

Summary

• Newton's First Law Every object will remain at rest or in uniform motion in a straight line unless it is made to change itsstate by the action of an unbalanced force .
• Newton's Second Law The resultant force acting on a body will cause the body to accelerate in the direction of theresultant force The acceleration of the body is directly proportional to the magnitude of the resultant force and inversely proportional to the mass of the object.
• Newton's Third Law If body A exerts a force on body B then body B will exert an equal but opposite force on body A.
• Newton's Law of Universal Gravitation Every body in the universe exerts a force on every other body. The force is directly proportional to the product of the masses of the bodies and inversely proportional to the square of the distance between them.
• Equilibrium Objects at rest or moving with constant velocity are in equilibrium and have a zero resultant force .
• Equilibrant The equilibrant of any number of forces is the single force required to produce equilibrium.
• Triangle Law for Forces in Equilibrium Three forces in equilibrium can be represented in magnitude and direction by thethree sides of a triangle taken in order.
• Momentum The momentum of an object is defined as its mass multiplied by its velocity.
• Momentum of a System The total momentum of a system is the sum of the momenta of each of the objects in the system.
• Principle of Conservation of Linear Momentum: `The total linear momentum of an isolated system is constant' or `In an isolatedsystem the total momentum before a collision (or explosion) is equal to the total momentum after the collision (or explosion)'.
• Law of Momentum: The applied resultant force acting on an object is equal to the rate of change of the object's momentum andthis force is in the direction of the change in momentum.

End of chapter exercises

Forces and newton's laws

1. [SC 2003/11] A constant, resultant force acts on a body which can move freely in a straight line. Which physical quantity will remain constant?
   1. acceleration
   2. velocity
   3. momentum
   4. kinetic energy
2. [SC 2005/11 SG1] Two forces, 10 N and 15 N, act at an angle at the same point.

   

   Which of the following cannot be the resultant of these two forces?
   1. A 2 N
   2. B 5 N
   3. C 8 N
   4. D 20 N
3. A concrete block weighing 250 N is at rest on an inclined surface at an angle of 20 ${}^{\circ }$ . The magnitude of the normal force, in newtons, is
   • A 250
   • B 250 cos 20 ${}^{\circ }$
   • C 250 sin 20 ${}^{\circ }$
   • D 2500 cos 20 ${}^{\circ }$
4. A 30 kg box sits on a flat frictionless surface. Two forces of 200 N each are applied to the box as shown in the diagram. Which statement best describes the motion of the box?
   1. A The box is lifted off the surface.
   2. B The box moves to the right.
   3. C The box does not move.
   4. D The box moves to the left.

   

5. A concrete block weighing 200 N is at rest on an inclined surface at an angle of 20 ${}^{\circ }$ . The normal reaction, in newtons, is
   • A 200
   • B 200 cos 20 ${}^{\circ }$
   • C 200 sin 20 ${}^{\circ }$
   • D 2000 cos 20 ${}^{\circ }$
6. [SC 2003/11]A box, mass $m$ , is at rest on a rough horizontal surface. A force of constant magnitude $F$ is then applied on the box at an angle of 60 ${}^{\circ }$ to the horizontal, as shown.

   

   If the box has a uniform horizontal acceleration of magnitude, $a$ , the frictional force acting on the box is $...$
   1. A $Fcos{60}^{\circ }-ma$ in the direction of A
   2. B $Fcos{60}^{\circ }-ma$ in the direction of B
   3. C $Fsin{60}^{\circ }-ma$ in the direction of A
   4. D $Fsin{60}^{\circ }-ma$ in the direction of B
7. [SC 2002/11 SG] Thabo stands in a train carriage which is moving eastwards. The train suddenly brakes. Thabo continues to move eastwards due to the effect of
   1. A his inertia.
   2. B the inertia of the train.
   3. C the braking force on him.
   4. D a resultant force acting on him.
8. [SC 2002/11 HG1] A body slides down a frictionless inclined plane. Which one of the following physical quantities will remain constant throughout the motion?
   1. A velocity
   2. B momentum
   3. C acceleration
   4. D kinetic energy
9. [SC 2002/11 HG1] A body moving at a CONSTANT VELOCITY on a horizontal plane, has a number of unequal forces acting on it. Which one of the following statements is TRUE?
   1. A At least two of the forces must be acting in the same direction.
   2. B The resultant of the forces is zero.
   3. C Friction between the body and the plane causes a resultant force.
   4. D The vector sum of the forces causes a resultant force which acts in the direction of motion.
10. [IEB 2005/11 HG] Two masses of $m$ and $2m$ respectively are connected by an elastic band on a frictionless surface. The masses are pulled in opposite directions by two forces each of magnitude $F$ , stretching the elastic band and holding the masses stationary.

    

    Which of the following gives the magnitude of the tension in the elastic band?
    1. A zero
    2. B $\frac{1}{2}F$
    3. C $F$
    4. D $2F$
11. [IEB 2005/11 HG] A rocket takes off from its launching pad, accelerating up into the air.

    

    The rocket accelerates because the magnitude of the upward force, $F$ is greater than the magnitude of the rocket's weight, $W$ . Which of the following statements best describes how force $F$ arises?
    1. A $F$ is the force of the air acting on the base of the rocket.
    2. B $F$ is the force of the rocket's gas jet pushing down on the air.
    3. C $F$ is the force of the rocket's gas jet pushing down on the ground.
    4. D $F$ is the reaction to the force that the rocket exerts on the gases which escape out through the tail nozzle.
12. [SC 2001/11 HG1] A box of mass 20 kg rests on a smooth horizontal surface. What will happen to the box if two forces each of magnitude 200 N are applied simultaneously to the box as shown in the diagram.

    

    The box will ...
    1. A be lifted off the surface.
    2. B move to the left.
    3. C move to the right.
    4. D remain at rest.
13. [SC 2001/11 HG1] A 2 kg mass is suspended from spring balance X, while a 3 kg mass is suspended from spring balance Y. Balance X is in turn suspended from the 3 kg mass. Ignore the weights of the two spring balances.

    

    The readings (in N) on balances X and Y are as follows:

    	X	Y
    (A)	20	30
    (B)	20	50
    (C)	25	25
    (D)	50	50

14. [SC 2002/03 HG1] $P$ and $Q$ are two forces of equal magnitude applied simultaneously to a body at X.

    

    As the angle $\theta$ between the forces is decreased from 180 ${}^{\circ }$ to 0 ${}^{\circ }$ , the magnitude of the resultant of the two forces will
    1. A initially increase and then decrease.
    2. B initially decrease and then increase.
    3. C increase only.
    4. D decrease only.
15. [SC 2002/03 HG1] The graph below shows the velocity-time graph for a moving object:

    

    Which of the following graphs could best represent the relationship between the resultant force applied to the object and time?

    (a)	(b)	(c)	(d)

16. [SC 2002/03 HG1] Two blocks each of mass 8 kg are in contact with each other and are accelerated along a frictionless surface by a force of 80 N as shown in the diagram. The force which block Q will exert on block P is equal to ...

    

    1. A 0 N
    2. B 40 N
    3. C 60 N
    4. D 80 N
17. [SC 2002/03 HG1] Three 1 kg mass pieces are placed on top of a 2 kg trolley. When a force of magnitude $F$ is applied to the trolley, it experiences an acceleration $a$ .

    

    If one of the 1 kg mass pieces falls off while $F$ is still being applied, the trolley will accelerate at ...
    1. A $\frac{1}{5}a$
    2. B $\frac{4}{5}a$
    3. C $\frac{5}{4}a$
    4. D $5a$
18. [IEB 2004/11 HG1] A car moves along a horizontal road at constant velocity. Which of the following statements is true?
    1. A The car is not in equilibrium.
    2. B There are no forces acting on the car.
    3. C There is zero resultant force.
    4. D There is no frictional force.
19. [IEB 2004/11 HG1] A crane lifts a load vertically upwards at constant speed. The upward force exerted on the load is $F$ . Which of the following statements is correct?
    1. A The acceleration of the load is 9,8 m.s ${}^{-2}$ downwards.
    2. B The resultant force on the load is F.
    3. C The load has a weight equal in magnitude to F.
    4. D The forces of the crane on the load, and the weight of the load, are an example of a Newton's third law 'action-reaction' pair.
20. [IEB 2004/11 HG1] A body of mass $M$ is at rest on a smooth horizontal surface with two forces applied to it as in the diagram below. Force $F_1$ is equal to $Mg$ . The force $F_1$ is applied to the right at an angle $\theta$ to the horizontal, and a force of $F_2$ is applied horizontally to the left.

    

    How is the body affected when the angle $\theta$ is increased?
    1. A It remains at rest.
    2. B It lifts up off the surface, and accelerates towards the right.
    3. C It lifts up off the surface, and accelerates towards the left.
    4. D It accelerates to the left, moving along the smooth horizontal surface.
21. [IEB 2003/11 HG1] Which of the following statements correctly explains why a passenger in a car, who is not restrained by the seat belt, continues to move forward when the brakes are applied suddenly?
    1. A The braking force applied to the car exerts an equal and opposite force on the passenger.
    2. B A forward force (called inertia) acts on the passenger.
    3. C A resultant forward force acts on the passenger.
    4. D A zero resultant force acts on the passenger.
22. [IEB 2004/11 HG1] A rocket (mass 20 000 kg) accelerates from rest to 40 m $·$ s ${}^{-1}$ in the first 1,6 seconds of its journey upwards into space. The rocket's propulsion system consists of exhaust gases, which are pushed out of an outlet at its base.
    1. Explain, with reference to the appropriate law of Newton, how the escaping exhaust gases exert an upwards force (thrust) on the rocket.
    2. What is the magnitude of the total thrust exerted on the rocket during the first 1,6 s?
    3. An astronaut of mass 80 kg is carried in the space capsule. Determine the resultant force acting on him during the first 1,6 s.
    4. Explain why the astronaut, seated in his chair, feels “heavier” while the rocket is launched.
23. [IEB 2003/11 HG1 - Sports Car]
    1. State Newton's Second Law of Motion.
    2. A sports car (mass 1 000 kg) is able to accelerate uniformly from rest to 30 m $·$ s ${}^{-1}$ in a minimum time of 6 s.
       1. Calculate the magnitude of the acceleration of the car.
       2. What is the magnitude of the resultant force acting on the car during these 6 s?
    3. The magnitude of the force that the wheels of the vehicle exert on the road surface as it accelerates is 7500 N. What is the magnitude of the retarding forces acting on this car?
    4. By reference to a suitable Law of Motion, explain why a headrest is important in a car with such a rapid acceleration.
24. [IEB 2005/11 HG1] A child (mass 18 kg) is strapped in his car seat as the car moves to the right at constant velocity along a straight level road. A tool box rests on the seat beside him.

    

    The driver brakes suddenly, bringing the car rapidly to a halt. There is negligible friction between the car seat and the box.
    1. Draw a labelled free-body diagram of the forces acting on the child during the time that the car is being braked.
    2. Draw a labelled free-body diagram of the forces acting on the box during the time that the car is being braked.
    3. What is the rate of change of the child's momentum as the car is braked to a standstill from a speed of 72 km.h ${}^{-1}$ in 4 s. Modern cars are designed with safety features (besides seat belts) to protect drivers and passengers during collisions e.g. the crumple zones on the car's body. Rather than remaining rigid during a collision, the crumple zones allow the car's body to collapse steadily.
    4. State Newton's second law of motion.
    5. Explain how the crumple zone on a car reduces the force of impact on it during a collision.
25. [SC 2003/11 HG1]The total mass of a lift together with its load is 1 200 kg. It is moving downwards at a constant velocity of 9 m $·$ s ${}^{-1}$ .

    

    1. What will be the magnitude of the force exerted by the cable on the lift while it is moving downwards at constant velocity? Give an explanation for your answer. The lift is now uniformly brought to rest over a distance of 18 m.
    2. Calculate the magnitude of the acceleration of the lift.
    3. Calculate the magnitude of the force exerted by the cable while the lift is being brought to rest.
26. A driving force of 800 N acts on a car of mass 600 kg.
    1. Calculate the car's acceleration.
    2. Calculate the car's speed after 20 s.
    3. Calculate the new acceleration if a frictional force of 50 N starts to act on the car after 20 s.
    4. Calculate the speed of the car after another 20 s (i.e. a total of 40 s after the start).
27. [IEB 2002/11 HG1 - A Crate on an Inclined Plane] Elephants are being moved from the Kruger National Park to the Eastern Cape. They are loaded into crates that are pulled up a ramp (an inclined plane) on frictionless rollers.The diagram shows a crate being held stationary on the ramp by means of a rope parallel to the ramp. The tension in the rope is 5 000 N.

    

    1. Explain how one can deduce the following: “The forces acting on the crate are in equilibrium”.
    2. Draw a labelled free-body diagram of the forces acting on the crane and elephant. (Regard the crate and elephant as one object, and represent them as a dot. Also show the relevant angles between the forces.)
    3. The crate has a mass of 800 kg. Determine the mass of the elephant.
    4. The crate is now pulled up the ramp at a constant speed. How does the crate being pulled up the ramp at a constant speed affect the forces acting on the crate and elephant? Justify your answer, mentioning any law or principle that applies to this situation.
28. [IEB 2002/11 HG1 - Car in Tow] Car A is towing Car B with a light tow rope. The cars move along a straight, horizontal road.
    1. Write down a statement of Newton's Second Law of Motion (in words).
    2. As they start off, Car A exerts a forwards force of 600 N at its end of the tow rope. The force of friction on Car B when it starts to move is 200 N. The mass of Car B is 1 200 kg. Calculate the acceleration of Car B.
    3. After a while, the cars travel at constant velocity. The force exerted on the tow rope is now 300 N while the force of friction on Car B increases. What is the magnitude and direction of the force of friction on Car B now?
    4. Towing with a rope is very dangerous. A solid bar should be used in preference to a tow rope. This is especially true should Car A suddenly apply brakes. What would be the advantage of the solid bar over the tow rope in such a situation?
    5. The mass of Car A is also 1 200 kg. Car A and Car B are now joined by a solid tow bar and the total braking force is 9 600 N. Over what distance could the cars stop from a velocity of 20 m $·$ s ${}^{-1}$ ?
29. [IEB 2001/11 HG1] - Testing the Brakes of a Car A braking test is carried out on a car travelling at 20 m $·$ s ${}^{-1}$ . A braking distance of 30 m is measured when a braking force of 6 000 N is applied to stop the car.
    1. Calculate the acceleration of the car when a braking force of 6 000 N is applied.
    2. Show that the mass of this car is 900 kg.
    3. How long (in s) does it take for this car to stop from 20 m $·$ s ${}^{-1}$ under the braking action described above?
    4. A trailer of mass 600 kg is attached to the car and the braking test is repeated from 20 m $·$ s ${}^{-1}$ using the same braking force of 6 000 N. How much longer will it take to stop the car with the trailer in tow?
30. [IEB 2001/11 HG1] A rocket takes off from its launching pad, accelerating up into the air. Which of the following statements best describes the reason for the upward acceleration of the rocket?
    1. A The force that the atmosphere (air) exerts underneath the rocket is greater than the weight of the rocket.
    2. B The force that the ground exerts on the rocket is greater than the weight of the rocket.
    3. C The force that the rocket exerts on the escaping gases is less than the weight of the rocket.
    4. D The force that the escaping gases exerts on the rocket is greater than the weight of the rocket.
31. [IEB 2005/11 HG] A box is held stationary on a smooth plane that is inclined at angle $\theta$ to the horizontal.

    

    $F$ is the force exerted by a rope on the box. $w$ is the weight of the box and $N$ is the normal force of the plane on the box. Which of the following statements is correct?
    1. A $tan\theta =\frac{F}{w}$
    2. B $tan\theta =\frac{F}{N}$
    3. C $cos\theta =\frac{F}{w}$
    4. D $sin\theta =\frac{N}{w}$
32. [SC 2001/11 HG1] As a result of three forces $F_1$ , $F_2$ and $F_3$ acting on it, an object at point P is in equilibrium.

    

    Which of the following statements is not true with reference to the three forces?
    1. The resultant of forces $F_1$ , $F_2$ and $F_3$ is zero.
    2. Forces $F_1$ , $F_2$ and $F_3$ lie in the same plane.
    3. Forces $F_3$ is the resultant of forces $F_1$ and $F_2$ .
    4. The sum of the components of all the forces in any chosen direction is zero.
33. A block of mass M is held stationary by a rope of negligible mass. The block rests on a frictionless plane which is inclined at ${30}^{\circ }$ to the horizontal.

    

    1. Draw a labelled force diagram which shows all the forces acting on the block.
    2. Resolve the force due to gravity into components that are parallel and perpendicular to the plane.
    3. Calculate the weight of the block when the force in the rope is 8N.
34. [SC 2003/11] A heavy box, mass $m$ , is lifted by means of a rope R which passes over a pulley fixed to a pole. A second rope S, tied to rope R at point P, exerts a horizontal force and pulls the box to the right. After lifting the box to a certain height, the box is held stationary as shown in the sketch below. Ignore the masses of the ropes. The tension in rope R is 5 850 N.

    

    1. Draw a diagram (with labels) of all the forces acting at the point P, when P is in equilibrium.
    2. By resolving the force exerted by rope R into components, calculate the $...$
       1. magnitude of the force exerted by rope S.
       2. mass, m, of the box.
    3. Will the tension in rope R, increase, decrease or remain the same if rope S is pulled further to the right (the length of rope R remains the same)? Give a reason for your choice.
35. A tow truck attempts to tow a broken down car of mass 400 kg. The coefficient of static friction is 0,60 and the coefficient of kinetic (dynamic) friction is 0,4. A rope connects the tow truck to the car. Calculate the force required:
    1. to just move the car if the rope is parallel to the road.
    2. to keep the car moving at constant speed if the rope is parallel to the road.
    3. to just move the car if the rope makes an angle of 30 ${}^{\circ }$ to the road.
    4. to keep the car moving at constant speed if the rope makes an angle of 30 ${}^{\circ }$ to the road.
36. A crate weighing 2000 N is to be lowered at constant speed down skids 4 m long, from a truck 2 m high.
    1. If the coefficient of sliding friction between the crate and the skids is 0,30, will the crate need to be pulled down or held back?
    2. How great is the force needed parallel to the skids?
37. Block A in the figures below weighs 4 N and block B weighs 8 N. The coefficient of kinetic friction between all surfaces is 0,25. Find the force P necessary to drag block B to the left at constant speed if
    1. A rests on B and moves with it
    2. A is held at rest
    3. A and B are connected by a light flexible cord passing around a fixed frictionless pulley