Question

please help me with that​

Answer 1

Answer:

• Radius of Circle (r) is 10 m.

Given:

1. Inclined Angle (θ) = 45°
2. Speed of Cyclist (v) =36 Km/h = 10 m/s.
3. Acceleration due to gravity (g) = 10 m/s²

Explanation:

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From the Formula we Know,

$\large\bigstar \: {\boxed{\tt \tan(\theta) = \dfrac{v^2}{rg}}}$

$\bold{Here} \begin{cases}\text{v Denotes Velocity} \\ \text{r Denotes Radius} \\ \text{g Denotes Acceleration due to gravity} \\ \theta \text{ Denotes Angle of inclination (vertical)}\end{cases}$

Now,

$\large{\boxed{\tt \tan(\theta) = \dfrac{v^2}{rg}}}$

Substituting the values,

$\longmapsto \large{\tt \tan(\theta) = \dfrac{(10)^2}{r \times 10}}$

$\longmapsto \large{\tt \tan 45 \degree = \dfrac{(10)^2}{r \times 10}}$

$\longmapsto \large{\tt 1 = \dfrac{(10)^2}{r \times 10}}$

$\longmapsto \large{\tt r \times 10 = 100}$

$\longmapsto \large{\tt r = \dfrac{100}{10}}$

$\longmapsto \large{\tt r = \cancel{\dfrac{100}{10}}}$

$\longmapsto \large{\underline{\boxed{\red{\tt r = 10 \: m}}}}$

∴ Radius of Circle of the Cyclist (r) is 10 m.

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Extra Shots:

Deriving the Above Formula:-

#Refer the attachment for figure.

Here we Can see Ground Exerts A Normal Reaction " R " .

Resolving in Components we get R cosθ & R sinθ along Horizontal & vertical directions respectively.

Applying, Law of Equilibrium.

We get,

$\longmapsto \large{\tt R \sin \theta = \dfrac{Mv^2}{r} \: ---(1)}$

$\longmapsto \large{\tt R \cos \theta = Mg \: ---(2)}$

Dividing both Equations.

$\longmapsto \large{\tt \dfrac{R \sin \theta}{R \cos \theta} = \dfrac{Mv^2}{r \times Mg}}$

$\longmapsto \large{\tt \cancel{\dfrac{R \sin \theta}{R \cos \theta}} = \cancel{\dfrac{Mv^2}{r \times Mg}}}$

$\longmapsto \large{\tt \dfrac{ \sin \theta}{ \cos \theta} = \dfrac{v^2}{r \times g}}$

$\longmapsto \large{\underline{\boxed{\red{\tt \tan \theta = \dfrac{v^2}{rg}}}}}$

Hence Derived!

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