Verified answer

Question :

i) A hollow sphere of mass m and radius r is set into pure rolling on a rough horizontal surface with a speed v₀. It strikes a smooth wall elastically. Find speed and angular speed of the sphere after collision.

ii) What's is its velocity when it is pure rolling. Also find the loss in kinetic energy

Class : 11

Chapter : Rotational Dynamics

Topic : Rolling Motion

Formula Used :

[tex]\mathrm{Loss\:\: in\:\: Kinetic \:\:Energy\:\:=\:\:}\frac{1}{2}(mv_{o}^2)+\frac{1}{2}(I\omega_{o}^2)[/tex]

Solution :

i) Given that a hollow sphere of mass m and radius r is set into pure rolling on a rough horizontal surface with a speed v₀. It strikes a smooth wall elastically. We need to find the speed and angular speed of the sphere after collision.

• The rough horizontal surface does not effect as the sphere is already in pure rolling. If the sphere was not in pure rolling then friction effects the sphere. ( Refer to fig.1 )

• The forces acting on the sphere are N₂ , f₂ and N₁. Since N₂ and f₂ are non - impulsive so they do not impact.

• As N₁ is the only impulsive force acting here the angular momentum does not change. N₁ also passes through the centre so it does not have any torque. ( Refer to fig.2 - forces acting on the body )

• Angular momentum remains constant and linear velocity is v₀. After collision ( Refer to fig. 3) ⇒ So it is not in pure rolling

ii) To find the velocity of the sphere when it is in pure rolling

[tex]L_i\:=\:mv_oR-I_o\omega_o\\\\=mv_oR-\frac{2}{3}mR^2.\frac{v_o}{R}\\\\=\frac{1}{3}mv_0R\\\\\\L_f=mvR(1+\frac{2}{3})\\\\=\frac{5}{3}mvR\\\\v=\frac{v_o}{5}[/tex]

The velocity of the sphere when it is in pure rolling is v = v₀/5

iii) To find the loss in kinetic energy

[tex]\mathrm{Loss\:\: in\:\: Kinetic \:\:Energy\:\:=\:\:}\frac{1}{2}(mv_{o}^2)+\frac{1}{2}(I\omega_{o}^2)\\\\=\frac{1}{2}mv^2(1+\frac{2}{3})\\\\=\frac{1}{2}mv^2(v_o^2-v^2)(1+\frac{2}{3})\\\\=\frac{5}{6}mv_o^2(1-\frac{1}{25})\\\\=\frac{4}{5}mv_o^2\\[/tex]

Therefore the loss in Kinetic Energy is 4/5mv₀². This implies that 96% of the total Kinetic Energy is lost.

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