An example of mechanical force is the thrust of an airplane. The airplane's engines make use of a propulsion system, which creates a mechanical force or thrust. Without thrust, an aircraft cannot move in the air.
The airplane's engines do work on some gases, which are accelerated to the engine's rear, creating a propulsion force in the opposite direction or thrust. The magnitude of thrust is dependent on the quantity of accelerated gas and the velocity variance of the gases in the engine system.
Another example of mechanical force is a hammer. When a hammer is used to hit a nail into a wall, both the hammer's and the hammering person's mechanical force are used.