AP EAPCET 08-Jul-2022 Shift 1 Paper

Let at some extension x centre of sphere has a speed v and angular speed of rotation of sphere be ω. Then

Total energy of system,
E= Rotational Energy + Translational energy + Spring energy
⇒E=

1

2

Iω2+

1

2

Mv2+

1

2

kx2
for sphere, I=

2

5

MR2
Also no ship condition is,
v=Rω
So, total energy,
E=

1

2

Mv2+

1

2

⋅

2

5

MR2(

v

R

)2+

1

2

kx2
As total energy remains constant,

dE

dt

=0
⇒

d

dx

{

1

2

Mv2+

1

5

Mv2+

1

2

kx2}=0
⇒

d

dt

(

7

10

Mv2+

1

2

kx2)=0
⇒

7

10

M⋅2v

dv

dt

+

1

2

k⋅2x

dx

dt

=0
⇒

7

5

Mva+

kx

2

v=0⇒a=

−5

7

k

M

⋅x
As, acceleration ∝ - displacement so, motion is SHM.
Hence, ω2=

5

7

k

M

Time period of oscillation,
⇒T=

2π

ω

=2π√

7

5

M

k