Question

A solid cylinder of mass 8 kg and radius 50 cm is
rolling down a plane inclined at an angle of 30° with
the horizontal. Calculate (i) force of friction,

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\therefore{\text{Force\:of\:friction=}}\frac{40}{3}N}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{ \underline \bold{Given :}} \\ : \implies \text{Mass(m) = 8 \: kg} \\ \\ : \implies \text{Radius(r) = 0.5 \: m} \\ \\ : \implies \text{Angle \: of \: inclination = 30 \degree} \\ \\ \red{ \underline \bold{To \: Find:}} \\ : \implies \text{Force \: of \: friction = ? }$

• According to given question :

$: \implies \text{Force= Mass }\times \text{Acceleration} \\ \\ : \implies mg sin \theta - fr = ma - - - - - (1) \\ \\ \bold{\circ \: \: Torque \: about \: friction \: force} \\ : \implies \tau = Iw \\ \\ \bold{\circ \: \: I = MOI \: of \: cylinder} \\ \bold{\circ \: \: w = angular \: mometum \: of \: cylinder} \\ \\ : \implies r \times fr = I \times w \\ \\ : \implies \frac{r \times fr}{I} = \frac{a}{r} \\ \\ : \implies fr = \frac{aI}{ {r}^{2} } - - - - - (2) \\ \\ \bold{ \circ \: \: \: \: Putting \: value \: of \: fr \: in \: (1)} \\ : \implies mg \: sin \theta - \frac{aI}{ {r}^{2} } = ma \\ \\ \bold{\circ \: \: \: \: MOI\: of \: cylinder = \frac{1}{2}m {r}^{2} } \\ \\ : \implies mg \:sin \theta - \frac{a \times m {r}^{2} }{2 {r}^{2} } = a \\ \\ : \implies a = \frac{2}{3} g \: sin \theta - - - - - (3) \\ \\ \bold{\circ \: \: \: \: putting \: value \: of \: a \: in \: (2)} \\ : \implies fr = \frac{aI}{ {r}^{2} } \\ \\ : \implies fr = \frac{2g \: sin \theta \times m {r}^{2} }{3 \times 2( {r})^{2} } \\ \\ : \implies fr = \frac{mg \:sin \theta}{3} \\ \\ : \implies fr = \frac{8 \times 10 \times sin 30 \degree}{3} \\ \\ : \implies fr = \frac{80}{3 \times 2} \\ \\ \green{:\implies fr = \frac{40}{3} N}$

Answer 2

Explanation:

A solid cylinder of mass 8 kg and radius 50 cm is

rolling down a plane inclined at an angle of 30° with

the horizontal. Calculate (i) force of friction,