Question

When a hollow sphere is rolling without slipping on a rough horizontal surface then the percentage of its total K.E which is Translational is

Answer 1

Solution :

⏭ Given:

✏ A hollow sphere is rolling without slipping on a rough horizotal surface.

⏭ To Find:

✏ The percentage of its total KE which is translation is

⏭ Concept:

✏ Total kinetic energy associated with rolling body (without slipping) is given by

$\bigstar \sf \: \red{KE_{net} = KE_{Translation} + KE_{Rotational}} \\ \\ \bigstar \: \underline{ \boxed{ \bold{ \sf{ \purple{ \large{ \large{KE_{net} = \dfrac{1}{2} M {v}^{2} { \huge{(}}1 + \dfrac{ {K}^{2} }{ {R}^{2} }{ \huge{) }}}}}}}}}$

⏭ Calculation:

$\implies \sf \: for \: hollow \: sphere \\ \\ \dag \sf \: I = M {K}^{2} = \dfrac{2}{3} M {R}^{2} \\ \\ \dag \: \boxed{\sf { \blue{ \dfrac{ {K}^{2} }{ {R}^{2} } = \dfrac{2}{3} }}} \: \circ$

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$\mapsto \sf \: \%(KE)_{Translation} = \dfrac{KE_{Translation}}{KE_{net}} \times 100 \\ \\ \mapsto \sf \: \%(KE)_{Translation} = \dfrac{ \cancel{\dfrac{1}{2} M {v}^{2} }}{ \cancel{\dfrac{1}{2} M {v}^{2}} (1 + \dfrac{ {K}^{2} }{ {R}^{2} } )} \times 100 \\ \\ \mapsto \sf \: \%(KE)_{Translation} = \dfrac{1}{1 + \dfrac{2}{3} } \times 100 \\ \\ \mapsto \sf \: \%(KE)_{Translation} = \frac{3}{5} \times 100 \\ \\ \mapsto \: \underline{ \boxed{ \bold{ \sf{ \orange{ \large{\%(KE)_{Translation} = 60\% \: of \: KE_{net}}}}}}} \: \green{ \bigstar}$

Answer 2

Solution:-

Let us consider a sphere of mass is M and radius is R.

★ Using formula:-

Momentum of innertia, I = ⅔ MR²

★ Point to remember:-

Here , the hollow sphere is rolling without slipping.

Therefore, v = Rω.

• Here, v is the speed of centre of mass and ω is the angular speed.

★ Formulas to be noted:-

→ Transitional kinetic energy = ½ Mv²

→ Rotational kinetic energy = ½ Iω²

★ ½ Iω²

½ × (⅔ MR²) × (v/R²) = ⅓ Mv²

→ Total kinetic energy = ½ Mv² + ⅓Mv² = 5/6 Mv²

★ To find % of transitional energy:-

% of transitional energy = ½ Mv²) / (5/6 Mv²) × 100 = 60%

Hence, the percentage of total kinetic energy which is transitional is 60%.