Aglass plate whose index of refraction is 1.64 is immersed in a liquid. the surface of the glass is inclined at an angle of 47° with the vertical. a horizontal ray in the glass is incident on the interface. when the liquid is a certain alcohol, the incident ray arrives at the interface at the critical angle. the index of refraction of the alcohol is closest to:

The frequency of vibrations of a vibrating violin string is given by f = 1 2l t ρ where l is the length of the string, t is its tension, and ρ is its linear density.† (a) find the rate of change of the frequency with respect to the following. (i) the length (when t and ρ are constant) (ii) the tension (when l and ρ are constant) (iii) the linear density (when l and t are constant) (b) the pitch of a note (how high or low the note sounds) is determined by the frequency f. (the higher the frequency, the higher the pitch.) use the signs of the derivatives in part (a) to determine what happens to the pitch of a note for the following. (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates df dl 0 and l is ⇒ f is ⇒ (ii) when the tension is increased by turning a tuning peg df dt 0 and t is ⇒ f is ⇒ (iii) when the linear density is increased by switching to another string df dρ 0 and ρ is ⇒ f is ⇒

What would be the radius of the earth if it had its actual mass but had the density of nuclei? express your answer to two significant figures and include the appropriate units.

Athird point charge q3 is now positioned halfway between q1 and q2. the net force on q2 now has a magnitude of f2,net = 5.861 n and points away from q1 and q3. what is the value (sign and magnitude) of the charge q3?