Question

An object with a mass of `2 kg` is acted on by two forces. The first is `F_1= <-3 N , -5 N>` and the second is `F_2 = < 6 N, 7 N>`. What is the object's rate and direction of acceleration?

Answer 1

Adding the vectors gives a magnitude of `3.6` `N`, and an angle of `34^o`.

Using Newton's Second Law:

`a=F/m=3.6/2=1.8` `ms^-2` at `34^o` to the positive x-axis.

Explanation:

First we find the resultant force acting on the object, by adding the components of the force vectors:

`F_1+F_2=<-3,-5>+<6,7>`
`=<(-3+6),(-5+7)> =<3,2>`

So the resultant force vector is 3 in the x direction and 2 in the y direction. We can use Pythagoras' theorem to find the magnitude of the resultant: `sqrt(3^2+2^2)=sqrt13~~3.6` `N`.

To find the direction, we use trig: 3 and 2 are the opposite and adjacent sides of a right-angled triangle, so:

`tan theta=(opp)/(adj)=2/3` so `theta ~~ 34^o`

Using Newton's Second Law, the acceleration is given by:

`a=F/m=3.6/2=1.8` `ms^-2` at `34^o` to the positive x-axis.