Hi there!
In order for a block to begin sliding, the force due to STATIC friction must be overcome.
In this instance, the following forces are acting on the block ALONG the axis of the incline.
Force due to gravity:
On an incline, the component of the force due to gravity contributing to the object's downward movement is equivalent to the horizontal (sine) component.
[tex]F_g = Mgsin\theta[/tex]
Force due to static friction:
The force due to friction is equivalent to the normal force multiplied by the coefficient of friction.
The normal force is the cosine component (perpendicular to the incline), so:
[tex]N = Mgcos\theta\\\\F_s = \mu_sMgcos\theta[/tex]
To find the minimum angle for the block to begin sliding, we can set the two forces equal to 0. They work in opposite directions (let down the incline be negative and up the incline be positive).
[tex]\Sigma F = F_s - F_g\\\\0 = F_s - F_g\\\\0 = \mu_sMgcos\theta - Mgsin\theta\\\\Mgsin\theta = \mu_sMgcos\theta[/tex]
Cancel out 'Mg' and rearrange to solve for theta.
[tex]sin\theta = \mu_scos\theta\\\\tan\theta = \mu_s\\\\\theta = tan^{-1}(\mu_s) = tan^{-1}(.45) = \boxed{24.228^o}[/tex]
