Question

a Motor cyclist (to be treated as a point mass)is to undertake horizontal circle inside the cylindrical Wall of a well of inner radius 4 m . coefficient of static friction between the Tyres and the wall is 0.4. calculate the minimum speed and frequency necessary to perform this stunt.(use g =10m/s^2)​

Answer 1

Topic :- Rotational Dynamics

$\maltese\:\underline{\sf AnsWer :}\: \maltese$

• Cylindrical Wall of a well of inner radius (r) = 4 m.
• Coefficient of static friction between the Tyres and the wall (μ) = 0.4
• Acceleration due to gravity (g) = 10 m/s²

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

$\longrightarrow\:\:\tt V_{minimum} = \sqrt{\dfrac{Rg}{\mu}}$

$\longrightarrow\:\:\tt V_{minimum} = \sqrt{\dfrac{4 \times 10}{0.4}}$

$\longrightarrow\:\:\tt V_{minimum} = \sqrt{\dfrac{40}{0.4}}$

$\longrightarrow\:\:\tt V_{minimum} = \sqrt{\dfrac{40 \times 10}{0.4 \times 10}}$

$\longrightarrow\:\:\tt V_{minimum} = \sqrt{\dfrac{400}{4}}$

$\longrightarrow\:\:\tt V_{minimum} = \sqrt{100}$

$\longrightarrow\:\: \underline{ \boxed{\tt V_{minimum} = 10 \: {ms}^{ - 1} }}$

Hence,the minimum speed of the motorcyclist will be 10 m/s. Now, let's find the frequency (n) necessary to perform the stunt.

$\dashrightarrow\:\tt v = r \omega$

Where, v is the linear velocity , r is the radius and ω is the angular speed/velocity.

But we know that, ω = 2πn, [Where n is the frequency]

$\dashrightarrow\:\tt v = r \times 2\pi n$

$\dashrightarrow\:\tt 10= 4 \times 2\pi n$

$\dashrightarrow\:\tt \dfrac{10}{4} = 2\pi n$

$\dashrightarrow\:\tt \dfrac{5}{2} = 2\pi n$

$\dashrightarrow\:\tt \dfrac{5}{2} = 2 \times 3.14 \times n$

$\dashrightarrow\:\tt \dfrac{5}{2} = 6.28\times n$

$\dashrightarrow\:\tt n = \dfrac{2.5}{6.28}$

$\dashrightarrow\:\tt n = 0.3980$

$\dashrightarrow\: \underline{ \boxed{\tt n \approx 0.4\:rev/sec}}$