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A::B::C::DSolution :

(a) Angular frequency <br> ` omega = sqrt((k)/(m))` or `omega = sqrt((500)/(2))` <br> or `omega = 15.81 rad//s` <br> (b) Equation of motion of the block (while elevator is accelerating) is, <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/DCP_VO2_C14_S01_046_S01.png" width="80%"> <br> ` kx - mg = ma = m(g)/(3)` <br> `:. x = (4mg)/(3k)` <br> `= ((4)(2)(10))/((3)(500)) = 0.053 m` <br> or `x = 5.3 cm` <br> (c) (i) In equilibrium when the elevator has zero acceleration, the equation of motion is, <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/DCP_VO2_C14_S01_046_S02.png" width="80%"> <br> `kx_(0) = mg` <br> or `x_(0) = (mg)/(k) = ((2)(10))/(500)` <br> `= 0.04m = 4cm` <br> `:.` Amplitude `A = x - x_(0) = 5.3 - 4.0` <br> ` = 1.3cm` <br> (ii) At time `t = 0`, block is at `x = - A`. Therefore, substituting `x = - A` and `t = 0` in equation, <br> ` x = Asin (omega t + phi)` <br> We get initial phase `phi = (3pi)/(2)` <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/DCP_VO2_C14_S01_046_S03.png" width="80%">.