Question

A point objects of mass `m` is slipping down on a smooth hemispherical body of mass `M` and radius `R`. The point object is tied to a wall with an ideal string as shown. At a certain instant, speed of the hemisphere is `v` and its acceleration is a. Then speed `v_(p)` and acceleration `a_(p)` of a particle has value (Assume all the surfaces in contact are frictionless)

A. `v_(P)=vsin60^(@)`
B. `v_(P)=v`
C. `a_(P)=a`
D. `a_(P)=sqrt([((v^(2))/(R ))+(a(sqrt(3))/(2))]^(2)+((a)/(2))^(2))`

Answer 1

`x+Rtheta=` constant
`(dx)/(dt)+(Rd theta)/(dt)=0`
`:. omega=(d theta)/(dt)=(v)/(R )`
`alpha=domega//dt=a//R`
At `theta=60^(@)`,
`:. v _(P)=v` and `a_(P)=sqrt(((asqrt(3))/(2)+(v^(2))/(R ))+((a)/(2))^(2))`