Electromagnetic waves carry energy as well as linear and angular momenta. When a light pulse is reflected from, transmitted through, or absorbed by a material medium, energy and momentum (both linear and angular) are generally exchanged, while the total amount of each entity remains intact. The extent of such exchanges between light and matter can be deduced, among other methods, with the aid of the Doppler shift phenomenon. The main focus of the present paper is on the transfer of angular momentum from a monochromatic light pulse to spinning objects such as a mirror, an absorptive dielectric, or a birefringent plate. The fact that individual photons of frequency ωo carry energy in the amount of ℏωo, where ℏ is Planck's reduced constant, enables one to relate the Doppler shift to the amount of energy exchanged. Under certain circumstances, the knowledge of exchanged energy leads directly to a determination of the momentum transferred from the photon to the material body, or vice versa.