XI DPP-26 CIRCULAR MOTION

A rubber band of length l has a stone of mass m tied to its one end. It is whirled with speed v so that the stone describes a horizontal circular path. The tension T in the rubber band is -

• zero

• mv² /l

• > (mv²)/l

• < mv² /l

A smooth wire is bent into a vertical circle of radius a . A bead P can slide smoothly on the wire. the circle is rotated about diameter AB as axis with a speed  $\omega$  as shown in figure. The bead P is at rest with respect to the circular ring in the position shown. Then  $\omega^2$ is equal to-

• 2g/a

•  $\frac{2g}{\left(\frac{a}{\sqrt{3}}\right)}$  
  

•  $\frac{g\sqrt{3}g}{a}$  

•  $\frac{2a}{\left(\frac{g}{\sqrt{3}}\right)}$  

A simple pendulum of length L and mass M is oscillating in a plane about a vertical line between angular limits – ϕ and + ϕ. For an angular displacement ϕ [ | θ | < ϕ], the tension in the string and the velocity of the bob are T and v respectively. The following relation holds good under the above conditions-

• T = Mg cos θ

• T cos θ = Mg

• T – Mg cos θ = Mv²/L

• T + Mg cos θ = Mv²/L

A car is moving with a speed of 30 m/sec on a circular path of radius 500 m. Its is increasing at the rate of 2 m/sec². What is the acceleration of the car ?

• 9.8 m/sec²

• 2.7 m/sec²

• 2.4 m/sec²

• 1.8 m/sec²

A point moves along a circle with velocity v = at where a = 0.5 m/sec². Then the total acceleration of the point at the moment when it covered (1/10) th of the circle after beginning of motion -

• 0.5 m/sec²

• 0.6 m/sec²

• 0.7 m/sec²

• 0.8 m/sec²

A solid body rotates about a stationary axis so that its angular velocity depends on the rotation angle  $\phi$   as  $\omega=\omega_0-k\phi$  , where  $\omega_0$   and k are positive constants. At the moment t = 0, the angle $\phi$   = 0. Find the time dependence of rotation angle -

• $K.\omega_0e^{-kt}$

• $\frac{\omega_0}{K}\left[e^{-kt}\right]$

• $\frac{\omega_0}{K}\left[1-e^{-k.t}\right]$

• $\frac{K}{\omega_0}\left[e^{-kt}-1\right]$

A heavy particle hanging from a fixed point by a light inextensible string of length l is projected horizontally with speed $\sqrt{\left(gl\right)}$   . Then the speed of the particle and the inclination of the string to the vertical at the instant of the motion when the tension in the string equal the weight of the particle-

•  $\sqrt{\frac{3l}{g}},\cos^{-1}\left(\frac{3}{2}\right)$  

•  $\sqrt{\frac{l\ g}{3}},\cos^{-1}\left(\frac{2}{3}\right)$  

•  $\sqrt{\frac{3g}{l}},\cos^{-1}\left(\frac{2}{3}\right)$  

•  $\sqrt{\frac{gl}{3}},\ \sin^{-1}\left(\frac{2}{3}\right)$  

A particle of mass m is attached to one end of a string of length l while the other end is fixed to a point h above the horizontal table, the particle is made to revolve in a circle on the table so as to make p revolutions per second. The maximum value of p if the particle is to be in contact with the table will be-

• $2p\sqrt{gh}$

• $\sqrt{\left(\frac{g}{h}\right)}$

• $2p\sqrt{\left(\frac{h}{g}\right)}$

• $\frac{1}{2\pi}\sqrt{\left(\frac{g}{h}\right)}$

A car is moving with a speed V on a road inclined at an angle $\theta$ in a circular arc of radius r, the minimum coefficent of friction so that the car does not slip away-

• $\frac{V^2}{rg}=\mu\ \tan\theta$

• $\mu=\frac{V^2}{rg}$

• $\frac{V^2\cos\theta-rg\sin\theta}{rg\cos\theta+V^2\sin\theta}$

• $\frac{V^2\cos\theta-rg\sin\theta}{rg\cos\theta-V^2\sin\theta}$

A mass of 2.9 kg, is suspended from a string of length 50 cm, and is at rest. Another body of mass 100 gm moving horizontally with a velocity of 150 m/sec, strikes and sticks to it. What is the tension in the string when it makes an angle of 60º with the vertical

• 153.3 N

• 135.3 N

• 513.3 N

• 351.3 N