Question

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Answer 1

$\underline{\underline{\sf{\qquad Given \qquad}}}$

Acceleration due to gravity = 10 m/s²

Radius = 4 m

Coefficient of friction = 0.4

$\underline{\underline{\sf{\qquad To \: Find \qquad}}}$

We have to find the minimum speed and and frequency necessary to perform this stunt.

$\underline{\underline{\sf{\qquad Solution \qquad}}}$

$\tiny\underline{\bigstar\:\frak{First \: of \: all \: we \: will \: find \: the \: minimum \: speed \: of \: the \: motorcyclist : }}$

$:\implies \sf V_{\tiny{min}} = \sqrt{\dfrac{rg}{\mu}}$

Where,

Vₘᵢₙ = minimum speed

r = radius

g = acceleration due to gravity

μ = coefficient of friction

$\tiny\underline{\bigstar\:\frak{Now,plug \: in \: the \: given \: values \: in \: above \: formula, \: we \: get : }}$

$:\implies \sf V_{\tiny{min}} = \sqrt{\dfrac{4 \times 10}{0.4}}$

$:\implies \sf V_{\tiny{min}} = \sqrt{\dfrac{40}{0.4}}$

$:\implies \sf V_{\tiny{min}} = \sqrt{100}$

$:\implies \underline{ \boxed{\sf V_{\tiny{min}} = 10 \: m/s }}$

$\therefore\:\underline{\textsf{The minimum speed of the motorcyclist is \textbf{10 m/s}}}.$

$\tiny\underline{\bigstar\:\frak{Now, Let's \: calculate \: the \: frequency \: necessary \: to \: perform \: this \: stunt : }}$

$:\implies \sf v = r \omega$

Where,

v is the velocity of motorcyclist

r is the radius

ω is the angular velocity of motorcyclist

$\tiny\underline{\bigstar\:\frak{Now,plug \: in \: the \: given \: values \: in \: above \: formula, \: we \: get : }}$

$:\implies \sf v = r \times 2\pi n$

$:\implies \sf n = \dfrac{v}{2\pi r}$

$:\implies \sf n = \dfrac{10}{2 \times \pi \times 4}$

$:\implies \sf n = \dfrac{10}{8\pi }$

$:\implies \sf n = \dfrac{5}{4\pi }$

$:\implies \sf n = \dfrac{5}{4 \times 3.142 }$

$:\implies \sf n = \dfrac{5}{12.568}$

$:\implies \sf n = 0.39783$

$:\implies \underline{ \boxed{ \sf n \approx 0.4 \: rev/sec}}$

$\therefore\:\underline{\textsf{The frequency necessary to perform this stunt is \textbf{0.4 rev/sec}}}.$

Answer 2

Answer:

Given:

A motor cyclist ( to be treated as a point mass) is to undertake horizontal circles inside the cylindrical wall of a well of inner radius 8m. (use g = 10 m/s²)

To find:

Find the coefficient of static friction between the tyres and the well when the minimum speed to perform this stunt is 12m/s  

Solution:

From given, we have,

The speed of the motor cyclist = v = 12 m/s^2

The radius of the cylindrical wall of a well = r = 8 m

We know that the necessary centripetal force is provided by the friction between the road and the tyres, so we have,

F = mv²/r = μmg

μ = v²/rg

μ = 12² / (8 × 10)

μ = 144 / 80

μ = 1.8

Therefore, the coefficient of static friction between the tyres and the well is 1.8

hope this helps you