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Questions and Answers for Topic #06

Topic 6 Concepts and Skills

Vocabulary, Variables and Equations

Review Questions

Application Questions

Word Problems

Topic #06 Concepts and Skills

Link to "Concepts and Skills"
 

Topic 6 Vocabulary

Scalar (Quantity): A quantity that has only magnitude; it has a number and a unit, but has no statement of direction.

Vector (Quantity): A quantity that has both magnitude and direction; it has a number, a unit and a direction.

Displacement: The vector quantity that defines the distance and direction between two positions.

Velocity: The vector quantity that defines the speed and direction of a moving object. Descriptive adjectives such as instantaneous, constant and average may be used along with this term.

Acceleration: The vector quantity that defines the rate at which the speed of an object changes as well as the direction in which the change occurs. Descriptive adjectives such as instantaneous, constant and average may be used along with this term.

Force: A push or pull exerted on an object having magnitude and direction; it may be either a contact or long range force.

Graphical Representation: An arrow or arrow-tipped line segment that symbolizes a vector quantity with a specified length and direction.

Algebraic Representation: Representation of a vector with an italicized letter in bold face type, which is often used in printed materials; in handwritten materials the algebraic representation of a vector is accomplished by writing the appropriate variable symbol with a forward facing arrow drawn above it.

Resultant (Vector): The sum of two or more vectors.

Equilibrant (Vector): A single, additional force that is exerted on a an object to produce equilibrium, which has the same magnitude as the resultant force, but is opposite in direction.

Vector Resolution: The process of breaking a vector into its (x and y) components. 

Vector Component: The x or y portion of a vector that has been broken into its components.

Vector Addition: The process of adding two or more vectors together; this can be done graphically or algebraically. This is never as simple a task as simple arithmetic addition except when the vectors are in the same direction such as both are on the x-axis.
 

Review Questions:Chapter 4
Reviewing Concepts

Questions:
Section 4.1
1. Describe how you would add two vectors graphically.
2. Which of the following actions is permissible when you are graphically adding one vector to another: move the vector, rotate the vector, change the vector’s length?
3. In your own words, write a clear definition of the resultant of two or more vectors.  Do not tell how to find it, but tell what it represents.
4. How is the resultant displacement affected when two displacement vectors are added in a different order?
5. Explain the method you would use to subtract two vectors graphically.
6. Explain the difference between these two symbols: A and A.

Section 4.2
7. Describe a coordinate system that would be suitable for dealing with a problem in which a ball is thrown up into the air.
8. If a coordinate system is set up such that the positive x-axis points in a direction 30deg above the horizontal, what should be the angel between the x-axis and the y-axis?  What should be the direction of the positive y-axis?
9. The Pythagorean theorem is usually written c^2 = a^2 + b^2.  If this relationship is used in vector addition, what do a, b, and c represent?
10. Using a coordinate system, how is the angle or direction of a vector determined with respect to the axes of the coordinate system?

 

Application Questions:Chapter 4
Applying Concepts

Questions

11. A vector drawn 15 mm long represents a velocity of 30 m/s. How long should you draw a vector to represent a velocity of 20 m/s?
12. A vector that is 1cm long represents a displacement of 5km.  How many kilometers are represented by a 3-cm vector drawn to the same scale?
13. What is the largest possible displacement resulting from two displacements with magnitudes 3m and 4m?  What is the smallest possible resultant?  Draw sketches to demonstrate your answers.
14. How does the resultant displacement change as the angle between two vectors increases from 0degrees to 180 degrees?
15. A and B are two sides of a right triangle.  If tan ? = A/B,
  a. which side of the triangle is longer if tan ? is greater than one?
  b. which side is longer if tan ? is less than one?
  c. what does it mean if tan ? is equal to one?
16. A car has a velocity of 50km/h in a direction 60 degrees north of east.  A coordinate system with the positive x-axis pointing east and a positive y-axis pointing north is chosen.  Which component of the velocity vector is larger, x or y?
17. Under what conditions can the Pythagorean theorem, rather than the Law of Cosines, be used to find the magnitude of a resultant vector?
18. A problem involves a car moving up a hill so a coordinate system is chosen with the positive x-axis parallel to the surface of the hill.  The problem also involves a stone that is dropped onto the car.  Sketch the problem and show the components of the velocity vector of the stone
(FIGURE 4-12)

 

Word Problems:Problem Set #6:  Follow the written directions unless indicated otherwise by your instructor.

GRAPHICAL SOLUTIONS:  Solve these problems by carefully drawing the vector information described and measuring the final answer using a protractor and ruler. Check answers using Trig. 

1.  After walking 11 km due north from camp, a hiker then walks 11 km due east.
 a.  What is the total distance walked by the hiker?
 b.  Determine the total displacement from the starting point. 

2.  A plane flying due north at 1.00x10^2 m/s is blown due west at 5.0x10^1 m/s by a strong wind. Find the plane's resultant velocity.

3.  A motorboat heads due east at 16 m/s across a river that flows due south at 9.0 m/s.
 a.  What is the resultant velocity (speed and distance) of the boat?
 b.  If the river is 136m wide, how long does it take the motorboat to reach the other side?
 c.  how far downstream is the boat when it reaches the other side of the river?

4.  While flying due west at 120 km/h, an airplane is blown due north at 45 km/h by the wind. What is the plane's resultant velocity?

5.  A salesperson leaves the office and drives 26 km due north along a straight highway.  A turn is made onto a highway that leads in a direction of 60.0 degrees.  The driver continues on the highway for a distance of 62 km and then stops.  What is the total displacement of the salesperson from the office?

6.  Two soccer players kick the ball at exactly the some time.  One player's foot exerts a force of 66 N north.  The other's foot exerts a force of 88 N east.  What is the magnitude and direction of the resultant force on the ball?

7.  Two forces of  62 N each act concurrently on a point P.  Determine the magnitude of the resultant force acting on point P when the angle between the forces is as follows:
 a. 0.0 degrees  b. 30.0 degrees c. 60.0 degrees d. 90.0 degrees
 e. 180.0 degrees

8.  In problem 7, what happens to the resultant of two forces as the angle between them increases?

9.  A weather team releases a weather balloon.  The balloon's buoyancy accelerates it straight up at 15 m/s^2.  A wind accelerates it horizontally at 6.5 m/s^2.  What is the magnitude and direction (with reference to the horizontal) of the resultant acceleration?

10.  What is the vector sum of a 65 N force acting due east and a 32 N force acting due west?

11.  A plane flies due north at 225 km/h.  A wind blows it due east at 55 km/h. What is the magnitude and direction of the plane's resultant velocity?

12.  A meteoroid passes between the moon and the earth. A gravitational force of 6.0x10^2 N pulls the meteoroid towards the moon. At the same time, a gravitational force of  4.8x10^2 N pulls it toward the earth.  The angle between the two forces is 130.0 degrees.  The moon's force acts perpendicularly to the meteoroids original path.  What is the resultant magnitude and direction of the force acting on the meteoroid?  State the direction in reference to the meteoroids original path.

MATHEMATICAL SOLUTION:  Solve the following problems mathematically using trigonometry. Include a labeled sketch of each problem's vector information.

13.  A 110 N force and a 55 N force act on point P.  The 110 N force acts due north.  The 55 N force acts due east. What is the magnitude and direction of the resultant force?

14.  A motorboat travels at 8.5 m/s.  It heads straight across a river 110 m wide.
 a.  If the water flows at the rate of 3.8 m/s, what is the boat's resultant velocity?
 b.  How much time does it take the boat to reach the opposite shore?

15.  A boat heads directly across a river 41 m wide at 3.8 m/s.  The current is flowing at 2.2 m/s.
 a.  What is the resultant velocity of the boat?
 b.  How much time does it take the boat to cross the river?
 c.  How far downstream is the boat when it reaches the other side?

16.  A 42 km/h wind blows in the direction 215 degrees.  What is the resultant velocity of a plane that flies a heading of 125 degrees at 152 km/h?

17.  Two 15 N forces act concurrently on point P.  FInd the magnitude of their resultant when the angle between them is 
 a.0.0 degrees  b.30.0 degrees  c.90.0 degrees  d.120.0 degrees
 e.180.0 degrees

18.  A boat travels at 3.8 m/s and heads straight across a river 240m wide.  The river flows at 1.6 m/s.
 a.  What is the boat's resultant speed with respect to the river bank?
 b.  How long does it take the boat to cross the river?
 c.  how far downstream is the boat when it reaches the other side?

19.  Determine the magnitude of the resultant of a 4.0x10^-1 N force and a 7.0x10^1 N force concurrently acting when the angle between them is 
 a.0.0 degrees b.30.0 degrees  c.60.0 degrees  d.90.0 degrees 
 e.180.0 degrees

SOLVING FOR THE EQUILIBRANT:  The equilibrant is a vector having exactly the same magnitude as the resultant vector, but is exactly 180o opposite the direction of the resultant.

20.  A force of 55N acts due west on an object.  What added single force on the object produces equilibrium?

21.  Two forces act concurrently on a point P.  One force is 6.0x10^1 N due east.  The second force is 8.0x10^1 N due north.
 a.  Find the magnitude and direction of the resultant
 b.  What is the magnitude and direction of their equilibriant?

22.  A 62N force acting at 30.0 degrees and a second 62N force acting at a 60.0 degrees are concurrent forces.
 a.  Determine the resultant force?
 b.  What is the magnitude and direction of their equilibrant?

23.  A  23 N force acts at 225 degrees.  A 48 N force acts at 315 degrees.  The two forces act on the same point.  What is the magnitude and direction of the equilibrant?

24.  A 33 N force acting due north and a 44 N force acting at 30 degrees act concurrently on a point P.  What is the magnitude and direction of a third force that produces equilibrium at point P?

RESOLVING A VECTOR INTO ITS COMPONENTS

25.  A heavy box is pulled across a wooden floor with a rope.  The rope forms an angle of 60 degrees with the floor.  A tension of 8.0x10^1 N is maintained on the rope.  What force actually is pulling the box across the floor?
 

26.  An airplane flies at 301 degrees at 5.0x10^2 km/h.  At what rate is the plane moving?
 a.  north.
 b. west.

27.  By applying a force of 72 N along the handle of a lawnmower, a student can push it across the lawn.  Find the horizontal component of this force when the handle is held at an angle with the lawn of 
 a. 60 degrees  b. 40.0 degrees c. 30.0 degrees

28.  A house address sign is hung from a post with a lightweight rod as shown in figure 6-14.  If the sign weighs 4.5 N, what is the force in the chain?

29.  A water skier is towed by a speedboat.  The skier moves to one side of the boat in such a way that the towrope forms an angle of 55 degrees with the wake of the boat.  The tension on the rope is 350 N.  What would be the tension on the rope if the skier were directly behind the boat?

GRAVITATIONAL FORCE AND INCLINED PLANES:  Using trig. to find the normal force.

30.  A 5.00x10^2 N trunk is placed on an inclined plane that forms a 66 degree angle with the horizontal.
 a.  Calculate the values of F perpendicular and F parallel.
 b.  Compare your results with those given for the some trunk at a 30 degree incline.
 c.  When the angle of an incline increases, how do the force components acting on the 
      trunk change?

31.  A car weighing 12000 N is parked on a 36 degree slope.
 a.  Find the force tending to cause the car to roll down the hill.
 b.  What is the perpendicular force between the car and the hill?

32.  In order to slide a 325 N trunk up a 20.0 degree inclined plane at a constant speed, a force of 211 N is applied.  What is the force of friction acting on the trunk?