$2.1.23.$ A body of mass $m$ lying on a horizontal plane is affected by a force $F$ at an angle $\alpha$ to the horizon. Coefficient of friction $\mu$. Find the acceleration of the body if it does not detach from the plane.
1. The normal reaction of the connection in this case will be determined both by the force of gravity $mg$ and the projection on the axis $OY$ of the applied force:
$N=mg-F \, sin \, \alpha$
The friction force is determined as:
$F_{тре} = (mg-F \,sin\, \alpha)$
2. The basic law of dynamics, therefore. will be written as follows:
$F \, cos \,\alpha = \mu (mg-F \,sin\, \alpha)$
3. From the equation of Newton's second law it is easy to determine the desired acceleration
$a = \frac{1}{m}(F\,cos\,\alpha-\mu mg+F\,sin\,\alpha)$
$a = \frac{F}{m}(cos\,\alpha-\mu \,sin\,\alpha)$
$a = (F/m)(cos \,α + \mu \,sin \, \varphi)$$ − \mu g$ if this expression is greater than zero, otherwise $a = 0$.