16.4 Components  (Page 3/3)

End of chapter exercises: vectors

1. An object is suspended by means of a light string. The sketch shows a horizontal force $F$ which pulls the object from the vertical position until it reaches an equilibrium position as shown. Which one of the following vector diagrams best represents all the forces acting on the object?

   

   A	B	C	D

2. A load of weight $W$ is suspended from two strings. $F_1$ and $F_2$ are the forces exerted by the strings on the load in the directions show in the figure above. Which one of the following equations is valid for this situation?
   1. $$ $W=F_1^2+F_2^2$
   2. $$ $F_{1}sin{50}^{\circ }=F_{2}sin{30}^{\circ }$
   3. $$ $F_{1}cos{50}^{\circ }=F_{2}cos{30}^{\circ }$
   4. $$ $W=F_1+F_2$

   

3. Two spring balances $P$ and $Q$ are connected by means of a piece of string to a wall as shown. A horizontal force of 100 N is exerted on spring balance Q. What will be the readings on spring balances $P$ and $Q$ ?

   

   	P	Q
   A	100 N	0 N
   B	25 N	75 N
   C	50 N	50 N
   D	100 N	100 N

4. A point is acted on by two forces in equilibrium. The forces
   1. have equal magnitudes and directions.
   2. have equal magnitudes but opposite directions.
   3. act perpendicular to each other.
   4. act in the same direction.
5. A point in equilibrium is acted on by three forces. Force $F_1$ has components 15 N due south and 13 N due west. What are the components of force $F_2$ ?
   1. 13 N due north and 20 due west
   2. 13 N due north and 13 N due west
   3. 15 N due north and 7 N due west
   4. 15 N due north and 13 N due east

   

6. Which of the following contains two vectors and a scalar?
   1. distance, acceleration, speed
   2. displacement, velocity, acceleration
   3. distance, mass, speed
   4. displacement, speed, velocity
7. Two vectors act on the same point. What should the angle between them be so that a maximum resultant is obtained?
   1. $0{}^{\circ }$
   2. $90{}^{\circ }$
   3. $180{}^{\circ }$
   4. cannot tell
8. Two forces, 4 N and 11 N, act on a point. Which one of the following cannot be the magnitude of a resultant?
   1. 4 N
   2. 7 N
   3. 11 N
   4. 15 N
   
   

End of chapter exercises: vectors - long questions

1. A helicopter flies due east with an air speed of $150\mathrm{km}·\mathrm{h}{}^{-1}$ . It flies through an air current which moves at $200\mathrm{km}·\mathrm{h}{}^{-1}$ north. Given this information, answer the following questions:
   1. In which direction does the helicopter fly?
   2. What is the ground speed of the helicopter?
   3. Calculate the ground distance covered in 40 minutes by the helicopter.
2. A plane must fly 70 km due north. A cross wind is blowing to the west at $30\mathrm{km}·\mathrm{h}{}^{-1}$ . In which direction must the pilot steer if the plane flies at a speed of $200\mathrm{km}·\mathrm{h}{}^{-1}$ in windless conditions?
   
3. A stream that is 280 m wide flows along its banks with a velocity of $\mathrm{1,80}\mathrm{m}·\mathrm{s}{}^{-1}$ . A raft can travel at a speed of $\mathrm{2,50}\mathrm{m}·\mathrm{s}{}^{-1}$ across the stream. Answer the following questions:
   1. What is the shortest time in which the raft can cross the stream?
   2. How far does the raft drift downstream in that time?
   3. In what direction must the raft be steered against the current so that it crosses the stream perpendicular to its banks?
   4. How long does it take to cross the stream in part c?
4. A helicopter is flying from place $X$ to place $Y$ . $Y$ is 1 000 km away in a direction ${50}^{\circ }$ east of north and the pilot wishes to reach it in two hours. There is a wind of speed $150\mathrm{km}·\mathrm{h}{}^{-1}$ blowing from the northwest. Find, by accurate construction and measurement (with a scale of $1\mathrm{cm}=50{\mathrm{km}·\mathrm{h}}^{-1}$ ), the
   1. the direction in which the helicopter must fly, and
   2. the magnitude of the velocity required for it to reach its destination on time.
5. An aeroplane is flying towards a destination 300 km due south from its present position. There is a wind blowing from the north east at $120\mathrm{km}·\mathrm{h}{}^{-1}$ . The aeroplane needs to reach its destination in 30 minutes. Find, by accurate construction and measurement (with a scale of $1\mathrm{cm}=30{\mathrm{km}·\mathrm{s}}^{-1}$ ), or otherwise,
   1. the direction in which the aeroplane must fly and
   2. the speed which the aeroplane must maintain in order to reach the destination on time.
   3. Confirm your answers in the previous 2 subquestions with calculations.
6. An object of weight $W$ is supported by two cables attached to the ceiling and wall as shown. The tensions in the two cables are $T_1$ and $T_2$ respectively. Tension $T_1=1200\mathrm{N}$ . Determine the tension $T_2$ and weight $W$ of the object by accurate construction and measurement or by calculation.

   

7. In a map-work exercise, hikers are required to walk from a tree marked A on the map to another tree marked B which lies 2,0 km due East of A. The hikers then walk in a straight line to a waterfall in position C which has components measured from B of 1,0 km E and 4,0 km N.
   1. Distinguish between quantities that are described as being vector and scalar .
   2. Draw a labeled displacement-vector diagram (not necessarily to scale) of the hikers' complete journey.
   3. What is the total distance walked by the hikers from their starting point at A to the waterfall C?
   4. What are the magnitude and bearing, to the nearest degree, of the displacement of the hikers from their starting point to the waterfall?
8. An object $X$ is supported by two strings, $A$ and $B$ , attached to the ceiling as shown in the sketch. Each of these strings can withstand a maximum force of 700 N. The weight of $X$ is increased gradually.
   1. Draw a rough sketch of the triangle of forces, and use it to explain which string will break first.
   2. Determine the maximum weight of $X$ which can be supported.

   

9. A rope is tied at two points which are 70 cm apart from each other, on the same horizontal line. The total length of rope is 1 m, and the maximum tension it can withstand in any part is 1000 N. Find the largest mass ( $m$ ), in kg, that can be carried at the midpoint of the rope, without breaking the rope. Include a vector diagram in your answer.