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Vectors

Force vector

Force is a vector. Supposing that two forces, one a 30 N force pointing north and another force, a 40 N force pointing east act simultaneously on a 1 kg block, as shown in Figure 7. We wish to know the block’s resultant acceleration.

Figure 7

We first add the two vectors to find the resultant force:

Figure 8

Move the first vector so that its tail coincides with the head of the other vector, join the tail of the second to the head of the first. See Figure 8.

The resultant is a vector 50 units long, acting in the direction shown (the arrow pointing diagonally to the right). The two forces acting together produce exactly the same effect on the object as the single resultant force. Applying Newton’s second law,

(2)

We conclude that the 1 kg block has an acceleration of 50 m/s2 in the direction of the resultant.

Q. A 2 N force pointing south and a 2 N force pointing east act simultaneously on a 2 kg block. What single force can balance the two forces so that the block stays in equilibrium?

A. A 2√2 N force acting toward the northwest direction.

Q. A 2 N force pointing southwest and a 2 N force pointing southeast act simultaneously on a 2 kg block for several seconds. If the block started at rest at the origin (of a coordinate system where north is along the positive y-axis and east is along the positive x-axis), what is its position at t = 4 sec?

A. The resultant force is 2√2 N pointing south. The acceleration is √2 m/s2. Using s = ½ a t2, its displacement is 8√2 m in the southern direction. So its position is (0, -8√2).

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