Maximum efficiency of a screw jack for the angle of friction $$\theta $$, is A. $$\frac{{\sin \theta }}{{1 + \sin \theta }}$$ B. $$\frac{{1 – \sin \theta }}{{\sin \theta }}$$ C. $$\frac{{1 + \sin \theta }}{{1 – \sin \theta }}$$ D. $$\frac{{1 – \sin \theta }}{{1 + \sin \theta }}$$

The correct answer is $\boxed{\frac{{1 – \sin \theta }}{{1 + \sin \theta }}}$.

The efficiency of a screw jack is the ratio of the output force to the input force. The output force is the force that is applied to the load, and the input force is the force that is applied to the screw. The angle of friction is the angle between the direction of the input force and the direction of the output force.

The maximum efficiency of a screw jack occurs when the angle of friction is equal to the angle of the threads on the screw. The angle of the threads on a screw is the angle between the helix of the screw and the horizontal.

The following equation can be used to calculate the efficiency of a screw jack:

$$\eta = \frac{{1 – \sin \theta }}{{1 + \sin \theta }}$$

where $\theta$ is the angle of friction.

The efficiency of a screw jack is always less than 1. This is because there is always some friction between the screw and the nut. The friction causes some of the input force to be lost as heat.

The efficiency of a screw jack can be improved by using a lubricant to reduce friction. The efficiency of a screw jack can also be improved by using a screw with a smaller angle of the threads.