An object and a screen are fixed on an optical bench and are separated by a distance L=90cm. A concave spherical mirror is moved along the bench. Two distinct mirror positions produce sharply focused, inverted images on the screen. The distance between these two mirror positions is measured to be D=30cm.
(a) Show that the focal length f of the mirror can be expressed in terms of L and D by the relation f=4LL2−D2.
(b) Using the measured values of L and D, calculate the focal length of the mirror.
(c) Determine the mirror’s radius of curvature.
(d) Derive an expression for the fractional uncertainty in f in terms of the fractional uncertainties in L and D. Discuss whether the method used in this experiment minimizes the impact of measurement uncertainties on the calculated focal length.