Question

Two uniform spheres of identical mass and radius are placed on inclined planes at the same height h and inclination angle θ. One plane is rough and causes one sphere to roll down the plane; the other is frictionless, and so the sphere on it slides down the incline.

(a) Find the ratio of the kinetic energies of the two spheres at the bottom of the incline:   .

Kslide Krell

(b) Find the ratio of their speeds at the base of the inclines: $\frac{v_{slide}}{v_{roll}}$ .

(c) Which sphere takes the greater time to reach the bottom of the incline or do they take the same time? Find the ratio of the times it takes the spheres to reach the bottom of the inclines: $\frac{t_{slide}}{t_{roll}}$ ​​​​​​​

Answer 1

B Ragh smooth frg- (Mass = m. Radin -R) Fry-2 @ ratio of KIE=? by energy conservation at figol migh = I m van + 1 Icnwoh 1 - не- | righ - I movent + 3 Mo Nen [ Vem - WP gh = Van + Vert - 7 van Puse roll Hence Totul KE = 1 Mvent I In w? = 5 mgh + 2 Mgh = mgh o The kinetic energy at bottom is = 5 Moru - © 7

Now for frg. 2 Mgu = Ivan (for no rotation as foretron absent) = 2gh Vonarson KE prey = 1 Mar - Imx2.gh = Mgh - Hence Mogh 5 Mgh - 7 2 7:5 Koll An 2 agh x 7 Voll logh = JF :55 AM © the acc. for fight (Pure rolling) is given by a g sino (utere. Kis (1tk) Gynection) a= g sim o rasin Sadius of Carecol frem fig dist s = ļg to s=h trevel) h Case ( 0 = 1 a troll -1 Саксо

7 roll - 124 Caseco V 5gsmo 14n Caneca 5gsmo - s Now for fig-2 s = a (az gaino onlly no rolling) h Careco a t at slide treide = ( 24 Care & - J 21 cm ② & ④ we can say in rolling is more by Comperreing eg that time taken then seiding. Aslide 14h Suche 241 = 57:55 A