Question

An object with a mass of `3 kg` is acted on by two forces. The first is `F_1= < -2 N , -3 N>` and the second is `F_2 = < -1 N, 6 N>`. What is the object's rate and direction of acceleration?

Answer 1

rate:`|vec a| = sqrt 2`

direction; `theta = (3pi) /4`

Explanation:

Newton's 2nd law

`Sigma vec F = m vec a`

`Sigma vec F = ((-2),(-3)) + ((-1),(6)) = ((-3),(3))`

`= m vec a = 3 vec a \qquad [= ((-3),(3))]`

So

`vec a = ((-1), (1))`

Breaking that vector into a scalar and a direction:

SCALAR:

`|vec a| = sqrt((-1)^2 + (1)^2) = sqrt 2`

DIRECTION:

in terms of direction, we can dot product it against the unit x vector , `hat x = ((1 ),(0))`, to get

`vec a * hat x`

`=((-1), (1)) * ((1 ),(0)) = -1 qquad triangle`

`= |vec a| |hat x| cos theta = |vec a| cos theta`

`= |(-1), (1)| cos theta`

`= sqrt 2 cos theta qquad square`

reconciling `square` and `triangle` means that

`cos theta= -1/ sqrt 2`

`theta = (3pi) /4`