Forces and equilibrium — common mistakes

In questions asking for a 'range of values' for a force to maintain equilibrium on a rough slope, you must consider two separate limiting cases: 1) The object is about to slide UP the slope (so friction acts DOWN). 2) The object is about to slide DOWN the slope (so friction acts UP). This will give you the maximum and minimum values for the required force.

Mass is a measure of the amount of matter in an object, measured in kilograms (kg). It is a scalar quantity and is the same everywhere. Weight is the force of gravity acting on an object's mass, measured in Newtons (N). It is a vector quantity ($W=mg$) and its value depends on the local gravitational field strength, $g$.

It depends on which angle you are using. A good rule is SOH CAH TOA. If you have a force F at an angle $\theta$ to the x-axis, the component adjacent to the angle $\theta$ (the x-component) is $F \cos\theta$. The component opposite the angle $\theta$ (the y-component) is $F \sin\theta$. Always draw a right-angled triangle with the force as the hypotenuse to be certain.

In mechanics, 'smooth' is a code word for 'frictionless'. It means you can assume the coefficient of friction is zero and there is no frictional force acting parallel to the surface. You still need to include the normal reaction force, which acts perpendicular to the surface.