Circular Motion on a Hemisphere: Finding Speed from Geometric Conditions

[IIK*JII]

Homework Statement

As shown in attached figure, a small object is in uniform circular motion in a horizontal plane, on the smooth of a hemisphere (radius:r). The distance between the object's plane of motion and the hemisphere's lowest point is \frac{r}{5}

What is the speed of the object?

Homework Equations

ƩF=\frac{mv²}{r} ...(1)

Ncosθ=mg ...(2)

The Attempt at a Solution

From (1)
and I get ƩF from FBD wrote in attached figure ƩF=Nsinθ

From (2) I knew that N=\frac{mg}{cosθ} ..(3)

substitute (3) in (1) in got gtanθ=\frac{v²}{r} ..(4)

and I try to find tanθ from geometric of hemisphere

First, I try to find the radius (let it is r') of this mass at r/5 from the lowest point of hemisphere

If I look in the picture and use pythagoras r' = (r²-(\frac{4r}{5}))^1/2
∴r' = \frac{3r}{5}
Thus; tanθ = 3
substitute in (4) v = √3gr

but the answer is \frac{3√5gr}{10}...

Or I get tanθ wrong or use wrong geometric condition of hemisphere ?

help is appreciate

Thanks :!)

 

[azizlwl]

Your tanθ should be equal to (3/5)r/(4/5)r=3/4

tanθ=v²/r'g
v²=3/4x3/5rg=3/4x3/5grx5/5=3.3.5gr/4.5.5

 

[IIK*JII]

Oh! thanks azizlwl

i get it this is an easy one but i can't notice