Trigonometry Method of Vector Addition



Trigonometry Method of Vector Addition

A way to add vectors in order to find the resultant R and its components vx and vy is to use trigonometry. The use of trig means that you don’t need a scale, don’t need a ruler, and don’t need a protractor! All you need is your calculator in order to evaluate sine and cosine.

Here’s what to do:

• Make and label a sketch of each vector.

If there are 2 vectors such that they and their resultant form a right triangle, then use

Pythagoras and Trig to find magnitudes and direction.

If the 2 vectors and their resultant don’t form a right triangle or there are more than

2 vectors, then:

• For each vector, find the horizontal component, vx, by using cos θ,

where θ is the angle made with the horizontal (or x-axis).

when using this method, we’ll generally call right (+x) and left (-x)

• Repeat the above step to find the vy for each vector using sin θ

• similarly, a common convention is to call up (+y) and down (-y)

• Add all of the individual vx’s to get the resultant vx.

• Add all of the individual vy’s to get the resultant vy.

• Use Pythagoras applied to the resultant components to find the magnitude of R.

• Use inverse trig functions to find and specify the direction of R.

Physics Trig Problems

1. Alex’s horse runs at 20 mph in a direction 58° S of E. Find its components.

(vx = 10.5 mph;

vy = 17 mph)

2. Scott’s car stalls. Find the components of a push applied if the push is 125 lb in a direction 30° N of E.

(vx = 108 lb;

vy = 62 lb)

3. TJ is bored and wants to go to the beach. He borrows Alex’s plane and tries to fly east. But a 50 mph southerly wind knocks the plane off course by 17°. Find the plane’s resultant speed and its original speed.

(170 mph, 163 mph)

4. Suppose the displacement between your home and WHHS is 4 miles in a direction 48° N of W. What are the components of your displacement from home when you are at school?

(vx = 2.7 mi ;

vy = 3 mi)

5. Sarah needs the plane to go skiing in Utah. She flies in a southwesterly direction and has a southern component of speed of 200 mph and a resultant speed of 350 mph. Find her western component of speed and exact direction.

(vx = 287 mph

55° W of S)

6. A boat tries to cross a stream at 10 m/s. The current downstream is 3 m/s. Find the resultant speed and direction of the boat.

(10.4 m/s , 17( S of E)

7. A 60 N force is directed at 30( N of E. A 30 N force is directed at 45( N of E.

Find the resultant force.

(89 N, 35( N of E)

8. Alex and Kelly go hiking at the Water Gap. Starting at the trailhead, they walk 10 miles at 53( S of W, then 7 miles at 20( N of W. Find their displacement from the trailhead.

(13.8 mi, 24( S of W)

9. Katey and Amy come to Scott’s aid. One of them pushes his car with 400 N at

30( N of W. Another pushes west with 340 N. The third pushes with 520 N at

60( S of W. Find their resultant force on the car.

(978 N, 15( S of W)

10. Marc’s boat has a top speed of 5 m/s in still water. He is motoring directly east across a river. The current of the river moves south at 3 m/s. Meanwhile a breeze is blowing at 4 m/s in a direction of 30( N of E. Find the boat’s resultant motion.

(8.5 m/s, 7( S of E)

11. Three forces act on an object to produce equilibrium, which means that the vector sum of the forces = 0. If one force is 33 N, north and the second force is 44 N at 60( N of E, find the 3rd force needed to produce equilibrium.

(note – this 3rd force is known as the equilibrant)

(74 N, 73( S of W)

12. Steven, Nora and Michelle push on a crate in such a way as to keep the crate in equilibrium. Steven pushes with a force of 50 N at 40( S of E. Nora pushes with a force of 80 N at 60( N of E. Find the force exerted by Michelle.

(86 N, 25( S of W)

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Special case for force vectors

When an object is in equilibrium, the force vectors acting on that object add up to zero (Σ F = 0).

Visually, this means that all of the force vectors meet head to tail…there 㐀㐖㐨㐪㐼㐾㑀㑄㑆㑊㑌㑐㑒㑖㑚㑜㒜㓈㓞㕒㕔㕨㕺is no room to draw a resultant!

Mathematically, this means that both the x components and the y components add up to zero!

3 m/s

10 m/s

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