Answer to Question #155276 in Physics for dat

Question

The horizontal and vertical components of a force are 12.0 N and 5.0 N respectively, When this force acts on a particle, the particle is displaced 3.2 m along the horizontal axis and 1.5 m along the vertical axis. Express (a) force and (b) displacement in unit vectors.  Find (c) the work done by this force on the particle. (d) Find the angle between force and the displacement

Answer 1

(a) The force in unit vector notation can be written as follows:

$$F=(12i+5j)\ N,$$

(b) The displacement in unit vector notation can be written as follows:

$$d=(3.2i+1.5j)\ m.$$

(c)-(d) The angle between force and the displacement vectors can be found as follows:

$$cos\theta=\dfrac{Fd}{||F||\cdot||d||},$$ $$\theta=cos^{-1}(\dfrac{Fd}{||F||\cdot||d||}),$$ $$\theta=cos^{-1}(\dfrac{(12i+5j)\cdot(3.2i+1.5j)}{\sqrt{12^2+5^2}\cdot \sqrt{3.2^2+1.5^2}}),$$ $$\theta=cos^{-1}(\dfrac{12\cdot3.2+5\cdot1.5}{13\cdot 3.53}),$$ $$\theta=2.4^{\circ}.$$

Finally, we can calculate the work done by this force on the particle:

$$W=Fdcos\theta=13\ N\cdot 3.53\ m\cdot 2.4^{\circ}=45.85\ J.$$

Answer:

(a) $(12i+5j)\ N$

(b) $d=(3.2i+1.5j)\ m$

(c)-(d) $\theta=2.4^{\circ}, W=45.85\ J.$