**JEE Main and Advanced**solved problems are two among those problems send to us by visitors of our blog as their queries. You can also send us your queries to us to get them solved.

**Problem 1.**Two masses m

_{1}and m

_{2}are connected by a spring of spring constant k and are placed on a friction less horizontal surface. Initially the spring is stretched through a distance x

_{0}when the system is released from rest. Find the distance moved by the messes before they again come to rest.

Solution.

Consider blocks plus spring a system and if no external forces acts on the system then centre of mass of system will remain at rest. Mean position of two SHM's would be the un stretched position.If m

_{1}moves towards right through a distance x

_{1}and m

_{2}moves towards left through a distance x

_{2 }before spring acquires natural length then

x

_{1}+ x

_{2}= x

_{0}.....................(1)

where x

_{1}and x

_{2}would be the amplitudes of blocks m

_{1}and m

_{2}resp. Since centre of mass of system will remain same so,

m

_{1}x

_{1}=m

_{2}x

_{2}

thus from 1

x

_{1}=m

_{2}x

_{0}/(m

_{1}+ m

_{2})

x

_{2}=m

_{1}x

_{0}/(m

_{1}+ m

_{2})

to get back to rest position masses m

_{1}and m

_{2}would have to travel distances x

_{1}and x

_{2}resp. , so total distance travelled by m

_{1}before comming to rest is

2m

_{2}x

_{0}/(m

_{1}+ m

_{2})

and total distance travelled by m

_{2}before comming to rest is,

2m

_{1}x

_{0}/(m

_{1}+ m

_{2})

**Problem 2.**Suppose the surface charge density over a sphere of radius R depends on a polar angle θ as σ=σ

_{0}cosθ, where σ

_{0}is a positive constant. Find the electric field strength vector inside the given sphere.

Solution.

You can visualize such a charge distribution as a result of a small relative shift of two uniformly charged balls of radius R, whose charges are equal in magnitude but opposite in sign. In such a representation inner charges cancel out each other and only charges on the surface survives having charge density σ=σ

_{0}cosθ. Charge density is maximum along +Z and -Z where the distance between the surfaces is maximum.

Electric field due to positively charged sphere

**E**=ρ

_{+}**r**/3ε

_{+}_{0}

Electric field due to positively charged sphere

**E**=-ρ

_{-}**r**/3ε

_{-}_{0}

Electric field at P

**E = E**=ρ(

_{+}+E_{-}**r**)/3ε

_{+}-r_{-}_{0}

or,

**E**=ρ

**a**/3ε

_{0}=ρakˆ/3ε

_{0}

now,

ρ

**a**= charge per unit area =σ

_{0}

hence,

**E**=σ

_{0}kˆ/3ε

_{0}

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