Question

A small object oscillates back and forth at the bottom of a frictionless hemispherical bowl, as the drawing illustrates. The radius of the bowl is R, and the angle is small enough that the object oscillates in simple harmonic motion. Derive an expression for the angular frequency w of the motion. Express your answer in terms of R and g, the magnitude of the acceleration due to gravity. Hemispherical bowl R

Answer 1

Solution! Let mass of object is m. radius of circe R free body diagram of object - - -- > masina ng inglösd & mg cosa ing where the N is the normal force apply by circle track on the object. = the centripital N-mglose will be balance force acting on the object H-mglosd = mer

force mgsind will be provied the dangential on the object. max = mg Sind Ax = g Sind s is very small than x will be also small => where & is the are displacement of object than dram =-07

is » negative sign indicate the dispacement oposite direction of acceleration x = R0 - (11) where R is the radius of circle & is the are length d is angle from the equation (11) (111) Ax = gsind

a is very small than Sino no an Ax=gd put the value of Art in equation (3) from equation (ll) and (W) R dd = -20 072 27 SHM equation Compair the equation (V) with SHM equation W²= a/R

W = Jala W = 8912 Ans