Lesson 8-5

Example 1
DOGS  Two children are attempting to capture a loose dog. One of the children is exerting a force of 80 Newtons due north and the other is pulling with a force of 100 Newtons due east. What is the resultant force on the dog?

a.	Draw a labeled diagram that represents the forces.

	Let  and  represent the forces exerted by	
the children. Then  represents the resultant.
Let  represent the angle  makes with the
east-west or x-axis.

b.	Determine the resultant force exerted on
the dog by the two children.

	||2	= ||2 + ||2
	||2	= (80)2 + (100)2
	||2	= 16,400
		=  or about 128.1

	The resultant force on the dog is about 128.1 Newtons.  1 pound is about 4.45 Newtons, so 128.1 N is about 28.8 pounds.

c.	Find the angle the resultant force makes with the east-west axis.

Use the tangent ratio.

	tan 	= 
		= tan-1 
		 38.7° north of due east

	The resultant force is applied at an angle of 38.7 north of due east.

Example 2
Sam works for a grocery store. Suppose he is pushing a cart full of produce weighing
75 pounds up a ramp 10 feet long at an incline of 20. Find the work done by gravity as
the cart moves the length of the ramp. Assume that friction is not a factor.

First draw a labeled diagram representing the	
forces involved. Let  represent the force
of gravity, or weight. The weight has a magnitude
of 75 pounds and its direction is down. The
corresponding unit vector is 0 - 75. So,
 = 0 - 75. The application of the force is ,
and it has a magnitude of 10 feet.

Write  as and use trigonometry to find x and y.

	cos 20°	= 
	x	= 10 cos 20°
	x	 9.40		sin 20°	= 
	y	= 10 sin 20°
	y	 3.42	

Then,  = 9.40 + 3.42.

Apply the formula for determining the work done by gravity.

W =   
W = 0 - 75  9.40 + 3.42
W = 0 - 256.5 or -256.5

Work done by gravity is negative when an object is lifted or raised. As the cart moves the length of the ramp, the work done by gravity is –256.5 ft-lb.

Example 3
Danielle is hanging a sign for her new	
book store. The sign is supported by
two lightweight support bars as shown
in the diagrams. If the bars make a 25
angle with each other, and the sign
weighs 150 pounds, what are the
magnitudes of the forces exerted by the
sign on each support bar?

 represents the force exerted on bar 1 by the sign,
 represents the force exerted on bar 2 by the sign,
and represents the weight of the sign.

Remember that equal vectors have the same	
magnitude and direction. So by drawing
another vector from the initial point of 
to the terminal point of , we can use
the sine and cosine ratios to determine || and ||.

	sin 25°	= 
	||	= 
		 354.93

	cos 25°	= 
	||	= 354.93 cos 25°
		 321.68

The sign exerts a force of about 322 pounds on bar 1 and a force of about 355 pounds on bar 2.

Example 4
A lighting system for a restaurant is supported equally by two cables suspended from the ceiling of the restaurant. The cables form a 150 angle with each other. If the lighting system weighs 800 pounds, what is the force exerted by each of the cables on the lighting system? 

Draw a diagram of the situation. Then draw the vectors tip-to-tail.

Since the triangle is isosceles, the base angles are congruent. Thus, each base angle measures   or 75. We can use the Law of Sines to find the force exerted by the cables.

		= 	Law of Sines
	x	= 
	x	 1545.48

The force exerted by each cable is about 1545 pounds.