1.) List all the vector quantities that we have studied so far in this course.

Displacement, velocity, acceleration and force are all vectors. They all have both a magnitude and a direction.














































2.) State two different methods that can be used to determine the components of any vector.

The components of any vector can be found both graphically (drawing them to scale on graph paper) or algebraically (using trigonometry).














































3.) State two different methods that can be used to add two or more vectors.

Two or more vectors can be added both graphically (drawing them to scale on graph paper) or algebraically (mostly using trigonometry). To add graphically, the tail of the next must be drawn at the tip of the previous. The resultant will always be drawn from the tail of the first to the tip of the last vector added. Algebraically, you must break down all the vectors into their components and then sum all the x-components and all the y-components separately. Using Pythagoras' theorem, the magnitude of the resultant can be found. Using trig, the angle at which it is acting can be found.














































4.) Can different vector quantities be added? (eg. force and velocity)

Different vector quantities such as force and velocity can not be added. It is much like trying to add apples and oranges, they are different so the result would not make any sense.














































5.) What happens when you change the order in which you add two or more vectors?

The order in which you add two or more vectors will NOT have any effect on the resultant. Similar to any addition, order is not important.














































6.) Two unequal forces are exerted on a box. Could the net force ever be zero?

If two unequal forces are exerted on a box the net force could never be zero. It is graphically impossible to take two directed line segments (vectors) of unequal length and place them tail to tip and end up with a resultant of zero. Similarly, you can not add two integers of unequal magnitude and end up with zero.














































7.) After adding two vectors, can the resultant vector be smaller than the smallest added? Explain.

After adding two vectors, the resultant vector could be smaller than the smallest added. To add vectors graphically, the tail of the next must be drawn at the tip of the previous. The resultant will always be drawn from the tail of the first to the tip of the last vector added. It is quite possible to draw two vectors using this rule and end up with a resultant vector smaller than any of those added. Similarly, this can be done when adding integers. The result could easily be smaller than any number being added.














































8.) Three unequal forces are exerted on a box. Could the net force ever be zero?

If three unequal forces are exerted on a box the net force could be zero. It is graphically possible to take three directed line segments (vectors) of unequal length and place them tail to tip and end up with a triangle. Therefore it ended at the same place that it started and the resultant is zero. Similarly, you can add three integers of unequal magnitude and end up with zero.














































9.) After adding three or more vectors, could the resultant
     vector ever be smaller than the smallest added? Explain.

After adding three vectors, the resultant vector could be smaller than the smallest added. To add vectors graphically, the tail of the next must be drawn at the tip of the previous. The resultant will always be drawn from the tail of the first to the tip of the last vector added. It is quite possible to draw three vectors using this rule and end up with a resultant vector smaller than any of those added. Similarly, this can be done when adding integers. The result could easily be smaller than any number being added.














































10.) Could one of the components of a resultant vector ever be smaller than the resultant?

Both of the components of a resultant vector MUST be smaller than the resultant. The resultant is the hypotenuse of a right angle triangle and the components are the opposite and adjacent sides. The hypotenuse is always the largest and therefore so is the resultant.














































11.) The components of a vector are perpendicular to each other.
        Explain how they can be considered independent of each other.

When we raced the cars down the chalk board it took a certain amount of time for them to go from top to bottom. The time it takes is not affected by moving the chalkboard horizontally. Similarly, if the car is not moving down the board while the board moves horizontally, the time it takes to go across the front of the class is the same whether the car is moving down or not. We also saw that if the car was moving down the board while the board moved across the room horizontally, that the resultant motion of the car could be described by a line that went at an angle from up high on the board to down low on the other corner. Therefore, we can conclude that motions that are perpendicular to each other can be considered independent of each other. And we can conclude that the two perpendicular components of any motion can be considered independent of each other.














































12.) You pull a toboggan on level ground at constant velocity by a rope at an angle to the horizontal.
        a) Can you increase the horizontal force on it without increasing the force you apply?
        b) Which component of the force applied is equal and opposite to the force of friction?
        c) If you stop pulling, what happens to the normal force?
        d) How could you decrease the force of friction against the toboggan?
        e) Would doing your answer to d) result in the toboggan speeding up?

a) The horizontal force on a toboggan being pulled at constant velocity with a rope that makes an angle with the horizontal can be increased by lowering the rope at you. By doing this, the rope makes a smaller angle with the horizontal and therefore the horizontal component of the force becomes larger while the vertical component becomes smaller. This can be proven by taking the cosine of two angles less than ninety degrees. This exercise will show that the smaller the angle the larger the cosine and therefore the larger the horizontal component.
b) The horizontal component of the force is the one that is equal and opposite to the force of friction. The force of friction acts in the opposite direction to the motion and since it is on level ground then it acts horizontally.
c) If you stop pulling, the normal force will get larger!! Since there is a component of the applied force in the vertical direction and it is pulling upward, then the force of gravity is counteracted by both this component and the normal force and not the normal force alone. The force of gravity doesn't change in either case so the normal is less when the applied force is acting.
d) You can decrease the force of friction against the toboggan by reducing the normal force (since the coefficient of friction is constant). To reduce the normal force you must have a larger vertical component of the applied force (see part c) To get a larger vertical component you must increase the angle that the rope makes with the horizontal (see part a).
e) Increasing the angle that the rope makes (holding the rope higher at your end) would NOT likely result in the toboggan speeding up. As you increase the angle, the force of friction goes down (see part d) but at the same time the horizontal component of the applied force gets smaller as well. And, the horizontal component would probably change more than the force of friction. So, most likely there will be a net force and it will slow down or it could possibly stay at constant velocity.














































13.) What is equilibrium?

Equilibrium is a state of balance. So, relative to forces, it is a special name for a net force of zero. All forces are balanced so the object is not accelerating (changing its present state of motion).














































14.) What is an equilibrant?

Equilibrium is a state of balance. The equilibrant is the special name given to a force that brings an object into a state of balance. In equilibrium the net force on an object is zero. So, for example, if the result of adding all the force vectors together is a net force other than zero, then the equilibrant is the force that would bring the net force to zero and therefore the object into equilibrium. With the equilibrant all the forces are balanced so the object would not accelerate (change its present state of motion).














































15.) Three forces are acting on the same object and it is in equilibrium. When two additional forces
        are applied, the object remains in equilibrium. What must be true about the additional forces?

Equilibrium is a state of balance. If three forces are acting on the same object and it is in equilibrium and then two additional forces are added and the object remains in equilibrium, then the two additional forces must have been equal and opposite.














































16.) Given a "sign problem", in which direction (vertical or horizontal) do we need to sum the forces?

In a "sign problem" we need to sum the the vertical forces. Usually there are no horizontally applied force to a sign so it is only the force of gravity in the vertical direction that needs to be considered.














































17.) In which of the two swings below is the rope more likely to break? Explain.
       

The ropes on the swing on the right are more likely to break because the ropes are at an angle. The ropes on the swing on the left are completely vertical and thus the force of gravity is divided evenly between the two ropes. So each rope needs to apply exactly half of the force of gravity pulling downward. The ropes on the swing on the right, on the other hand, are not completely vertical and thus the force of gravity is divided evenly between the two vertical components of the ropes. Since the rope provides the force, and any force is always larger than either of its components (see question #10), than each rope must apply a force greater than half the force of gravity.














































       

18.) In which of the two signs above (they have equal mass) is the wire more likely to break? Explain.

The wires on the picture on the right are more likely to break because the wires are at a smaller angle to the horizontal. The vertical components of the forces provided by the wires in both pictures must be equal and opposite to the force of gravity pulling downward on the picture. So each vertical component needs to apply exactly half of the force of gravity pulling down (it can be considered as two wires, one on each side of the nail). Because of the smaller angle, the force on the wires on the picture on the right (the hypotenuse) will need to be larger than the force on the wires on the picture on the left (the hypotenuse) to have the same size vertical components. This larger force is more likely to cause the wire to break.














































19.) Why is it almost impossible to stretch a string, rope, or wire perfectly
        horizontal when there is a small mass hanging in the middle ?

It is almost impossible to stretch a string, rope, or wire perfectly horizontal when there is a small mass hanging in the middle. When a string, rope or wire supports a mass it means that it must apply a force in the vertical direction that is equal and opposite to the force of gravity pulling down on the mass. When a string, rope or wire is perfectly horizontal, it has no vertical component and therefore can not support a mass. To support the mass then, the string, rope or wire must sag a little in the middle. The sag creates a slight angle at which the string, rope or wire is applying the force and therefore a small component of the force in the string, rope or wire acts vertically. The smaller the angle with the horizontal, the smaller the part of the force acts vertically. So, the applied force (straight along the line) that is required to support the mass gets greater and greater as the angle approaches zero (perfectly horizontal). Using machines, a large enough force can be created and as the wire begins to stretch (changes its properties) it can actually come very close to horizontal.














































20.) Why can you hang from a line that is vertical but not when it is strung horizontally?
         

You can hang from a line that is vertical but not when it is strung horizontally due the nature of vectors. When a vertically strung line supports itself or itself and you, all of the force needed to counteract the force of gravity is directed straight up the line. The applied force acts directly vertical. In this case, the force of gravity and the applied force are exactly the same. But, on the other hand, when a horizontally strung line supports a mass, the line must apply a force in the vertical direction that is equal and opposite to the force of gravity pulling down on the mass. When a line is perfectly horizontal, it has no vertical component and therefore can not support any mass. To support even its own mass then, the line must sag a little in the middle. The sag creates a slight angle at which the line is applying the force and therefore the force has a vertical component. Given a tightly strung horizontal line, even a small mass in the center will cause the line to sag more than without and it will cause an increase in the magnitude of the vertical component that is necessary to hold it up. Needing a larger vertical component in turn means needing an even greater force in the line (the hypotenuse). This, in many cases, will cause the line to break.














































21.) Why doesn't a ball start rolling when sitting on level ground?
        Make a drawing of the ball showing all forces.

A ball won't start rolling when sitting on level ground because there is not net force acting horizontally. Actually, threre are no horizontal forces at all. The only forces acting on the ball are acting vertically. Since the force of gravity is counterbalanced by the normal force, the ball stays in its present state of motion which, in this case, is sitting still. (a = 0)














































22.) Why will a ball start rolling when placed on a slope?
        Make a drawing of the ball showing all forces.

A ball will start rolling when placed on a slope because there is a net force acting down the slope. Because of the angle, there is a component of gravity acting parallel to the sloped surface. The component of the force of gravity that acts parallel to the slope is NOT counterbalanced by the force of friction acting up the slope. So, the ball does NOT stay in its present state of motion. It accelerates. The normal force will be equal and opposite to the component of gravity perpendicular to the slope. So, the ball will stay against the slope. There will be no acceleration in the direction perpendicular to the slope.














































23.) The skier shown (below left) is going at constant velocity down the hill.
        On the picture, draw all the forces acting on the skier.














































24.) The skier shown is accelerating down the hill.
        On the picture, draw all the forces acting on the skier.














































25. As the hill gets steeper:
      a) What happens to the force of gravity parallel to the hill?
      b) What happens to the force of gravity perpendicular to the hill?
      c) What happens to the normal force?
      d) What happens to the coefficient of friction?
      e) What happens to the force of friction?
      e) What happens to the acceleration?

a) The force of gravity parallel to the hill gets larger. Fgparallel = Fg sin θ. As the angle gets larger the sin of the angle also gets larger.

b) The force of gravity perpendicular to the hill gets smaller. Fgperpendicular = Fg cos θ. As the angle gets larger the cosine of the angle gets smaller.

c) The normal force gets smaller as the force of gravity perpendicular to the hill gets smaller.

d) The coefficient of friction is independent of the forces acting on the two surfaces in contact. It remains the same.

e) The force of friction will decrease as the normal force gets smaller.

f) The acceleration parallel to the slope would increase because the net force parallel to the slope gets larger. The net force is the sum of the force of gravity parallel to the slope and the force of friction. The force of gravity parallel to the slope gets larger and the force of friction gets smaller and since they are acting in opposite directions, the net force will get much larger causing a greater acceleration.














































26.) Newton's 2nd Law states: When the net force is zero, there is no change in velocity.(a=0)
       Why, then, do you have to pedal to keep a bike going at constant velocity on level ground
        but NOT when going down some hills?

You have to pedal to keep a bike going at constant velocity on level ground but NOT when going down a hill. On level ground, the force of friction (mostly air resistance) must be counterbalanced by an equal and opposite applied force. This applied force comes from the pedalling. On a hill, though, there is a component of gravity parallel to the slope. This component acts down the slope, the same direction that you are travelling. Since the net force must be zero to have constant velocity, the force of friction (mostly air resistance) must be counterbalanced by an opposing force or forces. In this case, you have stopped pedalling. So that applied force is now zero. But, the force of gravity parallel to the hill could be large enough to counterbalance the force of friction allowing you to continue on at a constant velocity.














































27.) Does a car burn more, the same, or less gas when it is going at constant
       velocity down a slight hill compared to on level ground? Explain.

A car will burn less gas when it is going at constant velocity down a hill compared to on level ground. You must use gas to keep a car going at constant velocity on level ground but NOT necessarily when going down a hill. On level ground, the force of friction (mostly air resistance) must be counterbalanced by an equal and opposite applied force. This applied force comes from the engine burning gas. On a hill, though, there is a component of gravity parallel to the slope. This component acts down the slope, the same direction that you are travelling. Since the net force must be zero to have constant velocity, the force of friction (mostly air resistance) must be counterbalanced by an opposing force or forces. The force of gravity parallel to the hill and the applied force from the engine are working together to counterbalance the force of friction and you could continue on at a constant velocity with a smaller applied force than on level ground and thus burn less gas.














































28.) A jet is flying at constant velocity during its ascent. The engines are creating 85 kN of thrust.
       What is the force of friction on this plane? Explain

If the engines of a jet are creating 85 kN of thrust during its ascent, the force of friction against the plane must be less than 85 kN. On a level flight path, the force of friction (air resistance) is the only force that must be counterbalanced by an equal and opposite applied force. This applied force comes from the jet engines. When a plane is climbing, though, there is a component of gravity parallel to the flight path which is making an angle with the horizontal. This component acts down the flight path (similar to a line parallel with the surface of a hill), in the opposite direction that the plane is travelling. Since the net force must be zero to have constant velocity, the force of friction (air resistance) and the component of gravity parallel to its path must be counterbalanced by the applied force from the engines. Therefore the force of friction must be less than the thrust created by the engines.














































29.) You are riding a bike on level ground at constant velocity. What happens to your speed
        if you keep pedalling with the same effort as you start to climb a hill? Explain.

If you are riding a bike on level ground at constant velocity and you start to climb a hill your speed will decrease if you keep pedalling with the same effort. On level ground, the force of friction (mostly air resistance) is the only force that must be counterbalanced by an equal and opposite applied force. This applied force comes from you pedalling. When you climb a hill, though, there is a component of gravity parallel to the hill. This component acts down the hill, in the opposite direction that you are pedalling. Since the net force must be zero to have constant velocity, the force of friction (mostly air resistance) and the component of gravity parallel to the hill must be counterbalanced by the applied force from pedalling to maintain constant velocity. But, since you continue to pedal with the same effort, the applied force stays the same and thus there is a net force acting down the slope. This net force means that there will be a change in velocity (an acceleration) and the bike will slow down.














































30.) In the previous question, if the slope was very gentle, would you slow down to a stop? Explain.

In the previous question, the bike would NOT slow down to a stop. Instead, the bike would slow down until the applied force was equal and opposite to the sum of the force of friction (air resistance) and the component of gravity acting parallel to the hill. Since the component of gravity acting parallel to the hill doesn't change, the bike would slow down to a speed where the air resistance was reduced by the magnitude of this parallel component of gravity.