Optics

Step 1: For a body moving in a horizontal circular path, the centripetal force required is provided by friction. Maximum frictional force:
Step 2: Centripetal force required: For no slipping:
Step 3: Cancel from both sides:
Step 4: Solve for angular velocity:
Step 5: Substitute the given values: Final Answer (up to two decimal places):

Step 1: Let the object distance from the mirror be cm and the image distance be cm. Given that the image is formed 10 cm farther from the mirror than the object:
Step 2: Using the mirror formula: For a concave mirror: Substitute:
Step 3: Simplifying: Solving:
Step 4: Then,
Step 5: Magnification for a mirror is: Final Answer (up to two decimal places):

Step 1: Identify the point of observation. The mid-point of the screen corresponds to the point where the path difference between the two waves is zero.
Step 2: Intensity at the mid-point for coherent sources. For two coherent waves of equal amplitude : Resultant amplitude: Since intensity is proportional to the square of amplitude:
Step 3: Intensity at the mid-point for incoherent sources. For incoherent sources, intensities simply add: Intensity due to one slit: Total intensity:
Step 4: Find the ratio of intensities. However, note that in Young’s experiment the maximum intensity at the center for coherent sources is four times the intensity due to one slit, whereas for incoherent sources it is equal to the sum of individual intensities. Thus:
Hence, the correct answer is .

Step 1: For triangular wave over full period:
Step 2: Any fraction of a linear ramp preserves same rms coefficient because Hence rms .

Step 1: Recall the concept of matter waves. According to de Broglie’s hypothesis, moving particles such as electrons exhibit wave-like properties. Step 2: Identify G. P. Thomson’s contribution. G. P. Thomson experimentally verified the wave nature of electrons by passing them through thin metal foils. Step 3: Observe the phenomenon involved. The electrons produced diffraction patterns, similar to X-ray diffraction, which is a characteristic property of waves. Step 4: Analyze the options.

(A) Diffraction — direct evidence of wave nature ✔
(B) Refraction — not observed for electrons
(C) Polarization — applies to transverse waves like light
(D) Scattering — not specific proof of wave nature
Step 5: Final conclusion. The phenomenon that confirmed the existence of matter waves is diffraction.

Step 1: Recall the wave nature of matter. According to de Broglie hypothesis, every moving particle has an associated wavelength: Step 2: Identify experimental evidence. The wave nature of matter is confirmed experimentally by diffraction and interference phenomena. Step 3: Analyze the options.

(A) Electron momentum — property of particles, not wave evidence
(B) Electron diffraction — direct evidence of wave nature ✔
(C) Photon momentum — photons are already waves/particles
(D) Photon diffraction — supports wave nature of light, not matter
Step 4: Final conclusion. Electron diffraction best supports the theory that matter has a wave nature.