(C) The mass $m$ moves in a horizontal circle of radius $r = \ell \sin \theta$. The angular frequency is $\omega = 2\pi f = 2\pi \times \frac{1}{\pi}

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(C) The mass $m$ moves in a horizontal circle of radius $r = \ell \sin \theta$. The angular frequency is $\omega = 2\pi f = 2\pi \times \frac{1}{\pi} = 2 \text{ rad/s}$. The forces acting on the mass are tension $T$ (along the string) and weight $mg$ (downwards). Resolving tension into components: $T \cos \theta = mg$ and $T \sin \theta = m\omega^2 r$. Substituting $r = \ell \sin \theta$,we get $T \sin \theta = m\omega^2 \ell \sin \theta$,which simplifies to $T = m\omega^2 \ell$. Substituting $\omega = 2 \text{ rad/s}$,we get $T = m(2)^2 \ell = 4m\ell$.

Class 11 Physics Newton's Laws of Motion and Friction Circular motion with Friction

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An inextensible string of length $\ell$ fixed at one end,carries a mass $m$ at the other end. If the string makes $\frac{1}{\pi}$ revolutions per second around the vertical axis through the fixed end,the tension in the string is [The string makes an angle $\theta$ with the vertical]

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A
$4\pi^2 m\ell$
B
$2\pi^2 m\ell$
C
$4 m\ell$
D
$m\ell$

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Newton's Laws of Motion and Friction

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Circular motion with Friction

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An inextensible string of length $\ell$ fixed at one end,carries a mass $m$ at the other end. If the string makes $\frac{1}{\pi}$ revolutions per second around the vertical axis through the fixed end,the tension in the string is [The string makes an angle $\theta$ with the vertical]

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$A$ particle moves on a circular overbridge with constant speed. The friction coefficient varies so that the speed remains constant. Which of the following graphs shows the magnitude of friction $f$ versus the angle $\theta$?

$A$ car has to move on a level turn of radius $450\,m.$ If the coefficient of static friction between the tyre and the road is $\mu = 0.2,$ find the maximum speed the car can take without skidding in $m/s.$

$A$ circular road of radius $1000 \, m$ has a banking angle of $45^\circ$. The maximum safe speed of a car having mass $2000 \, kg$ will be,if the coefficient of friction between the tyre and road is $0.5$,equal to ....... $m/s$.

$A$ van is moving with a speed of $108 \,km/hr$ on a level road where the coefficient of friction between the tyres and the road is $0.5$. For the safe driving of the van, the minimum radius of curvature of the road shall be (Acceleration due to gravity, $g = 10 \,m/s^2$) (in $\,m$)

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