1. Applications of Dot Product

A crate with a weight of 57 N rests on a frictionless ramp inclined at an angle of 30 degrees to the horizontal. What force must be applies at an angle of 20 degrees to the ramp so that the crate remains at rest?

I've drew the diagram and I don't know what to do after this:

I know the formula to use is "u (dot) v = |u||v|cosx"

The answer to this problem is 28.5 N and 30.3 N.

Can you please show me a step by step solution to get the answer?

2. Originally Posted by Macleef
A crate with a weight of 57 N rests on a frictionless ramp inclined at an angle of 30 degrees to the horizontal. What force must be applies at an angle of 20 degrees to the ramp so that the crate remains at rest?

I've drew the diagram and I don't know what to do after this:

I know the formula to use is "u (dot) v = |u||v|cosx"

The answer to this problem is 28.5 N and 30.3 N.

Can you please show me a step by step solution to get the answer?
I've attached a sketch with all the forces acting on the solid:

$\displaystyle F_w = \text{weight}$
$\displaystyle F_d = \text{downhill force}$
$\displaystyle F_u = \text{uphill force}$
$\displaystyle F_p = \text{force to pull}$

$\displaystyle | \overrightarrow{F_d} | =| \overrightarrow{ F_w} | \cdot \cos(60^\circ) = 28.5\ N$

$\displaystyle \overrightarrow{F_u} = -| \overrightarrow{ F_d}$

$\displaystyle |\overrightarrow{F_u}| = |\overrightarrow{F_p}| \cdot \cos(20^\circ)~\implies~ |\overrightarrow{F_p}| = \frac{|\overrightarrow{F_u}|}{\cos(20^\circ)}\appr ox 30.329\ N$

3. Originally Posted by earboth
I've attached a sketch with all the forces acting on the solid:

$\displaystyle F_w = \text{weight}$
$\displaystyle F_d = \text{downhill force}$
$\displaystyle F_u = \text{uphill force}$
$\displaystyle F_p = \text{force to pull}$

$\displaystyle | \overrightarrow{F_d} | =| \overrightarrow{ F_w} | \cdot \cos(60^\circ) = 28.5\ N$

$\displaystyle \overrightarrow{F_u} = -| \overrightarrow{ F_d}$

$\displaystyle |\overrightarrow{F_u}| = |\overrightarrow{F_p}| \cdot \cos(20^\circ)~\implies~ |\overrightarrow{F_p}| = \frac{|\overrightarrow{F_u}|}{\cos(20^\circ)}\appr ox 30.329\ N$
what is F_n?

4. Originally Posted by Jhevon
what is F_n?
sorry I forgot to mention:

$\displaystyle F_n = \text{normal force acting perpendicularly at the surface of the inclined plane}$