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question_answer1) Find the minimum velocity (in\[m{{s}^{-1}}\]) with which a car driver must travels on a flat curve of radius 150 m and coefficient of friction 0.6 to avoid skidding.
question_answer2) A wheel is subjected to uniform angular acceleration about its axis. Initially its angular velocity is zero. In the first 2 sec, it rotates through an angle\[{{\theta }_{1}}\]; in the next 2 sec, it rotates through an additional angle\[{{\theta }_{2}}\]. The ratio of \[{{\theta }_{2}}/{{\theta }_{1}}\]is?
question_answer3) A particle moves in a circle with a uniform speed, when it goes from a point A to a diametrically opposite point B, the momentum of the particle changes by \[{{\vec{P}}_{A}}-{{\vec{P}}_{B}}=2kg-m/s\left( {\hat{j}} \right)\]and the centripetal force acting on it changes by \[{{\vec{F}}_{A}}-{{\vec{F}}_{B}}=8N\left( {\hat{i}} \right)\] where \[\hat{i},\]\[\hat{j}\] are unit vectors. Find the angular velocity (in rad/s) of the particle.
question_answer4) A mass m is revolving in a vertical circle at the end of a string of length 20 cm. By how much does the tension (in mg) of the string at the lowest point exceed the tension at the top most point?
question_answer5) A boy whirls a stone in a horizontal circle of radius 1.5 m and 2 m above the ground by means of a string. The string breaks and the stone flies off horizontally, striking the ground 10 m away. The centripetal acceleration during circular motion is\[a\times \frac{490}{3}m/{{s}^{2}}\]. Determine a.
question_answer6) A pendulum of length L = 15 cm is held with its string horizontal and then released. The string runs into a peg a distance 'd' below the pivot, as shown in figure. What is the smallest value of 'd' (in cm) for which the string remains taut at all times?
question_answer7) A cube of mass M starts at rest from point 1 at a height 4R, where R is the radius of the circular track. The cube slides down the frictionless track and around the loop. What is the force (in N) which the track exerts on the cube at point 2?\[M=0.2\text{ }kg\].
question_answer8) A stone is thrown horizontally with a velocity 10 m/s. The radius of curvature of its trajectory in 3 second after the motion began is \[100\sqrt{2a}\]m. Disregard the resistance of air. Determine a.
question_answer9) A simple pendulum bob is suspended with the help of a string of length \[\ell \]. If at its equilibrium position, bob is imparted a velocity\[v=\sqrt{3g\ell },\] then the angle with vertical at which the pendulum bob will discontinue the loop is\[{{\tan }^{-1}}\]\[\left( {{x}_{0}}\sqrt{2} \right)\]. Find\[{{x}_{0}}\].
question_answer10) A particle of mass 0.2 kg moves along the internal smooth surface of vertical cylinder of radius 2m. Find the force in N with which the particle acts on the cylinder wall if at the initial moment of time its velocity equals 10 m/s and forms an angle \[45{}^\circ \]with the horizontal.
question_answer11) A heavy particle hanging from a fixed point by a light inextensible string of length \['\ell '\] is projected horizontally with speed\[\sqrt{g\ell }\], the speed of the particle is \[\sqrt{\frac{g\ell }{a}}\]and the inclination of the string to the vertical is \[{{\cos }^{-1}}\frac{2}{b}\]at the instant of the motion when the tension in the string is equal to the weight of the particle. Determine\[\frac{a}{b}\].
question_answer12) A table with smooth horizontal surface is fixed in a cabin that rotates with a uniform angular velocity \[\omega \]in a circular path of radius R. A smooth groove AB of length L<<R is made on the surface of the table. The groove makes an angle \[60{}^\circ \]with the radius OA of the circle in which the cabin rotates. A small particle is kept at the point A in the groove and is released to move along AB. The time taken by the particle to reach the point B is a\[\sqrt{\frac{L}{{{\omega }^{2}}R}}\]. Determine a.
question_answer13) A table with smooth horizontal surface is placed in a cabin which moves in a circle of a large radius R. A smooth pulley of small radius is fastened to the table. Two masses of m and 2m are placed on the table connected through a string going over the pulley. Initially the masses were at rest. The magnitude of the initial acceleration of the masses as seen from the cabin is \[{{a}_{rel}}\]and the tension in the string is T. Determine \[\frac{T}{{{a}_{rel}}}\].(m=1kg)
question_answer14) A smooth hemisphere of radius R is made to translate in a straight line with a constant acceleration. \[{{a}_{cc}}=g\]. A particle kept on the top of the sphere is released from there at zero velocity with respect to the sphere. The speed of the particle with respect to the sphere varies as a function of the angle \[\theta \] it slides when\[\theta =45{}^\circ \]. Speed is \[\sqrt{a\,gR}\]m/s. Determine a. (Assume particle does not leave the surface of sphere)
question_answer15) A massive horizontal platform is moving horizontally with a constant acceleration of \[10m/{{s}^{2}}\]as shown in the figure. A particle P of mass m = 1 kg is kept at rest at the smooth surface as shown in the figure. The particle is hinged at O with the help of a massless rod OP of length 0.9 m. Hinge O is fixed on the platform and the rod can freely rotate about O. Now the particle P is imparted a velocity in the opposite direction of the platform?s acceleration such that it is just able to complete the circular motion about point O. Then the maximum tension appearing in the rod during the motion is 10n. Find the value of n.
question_answer16) The figure shows the velocity and acceleration of a point like body at the initial moment of its motion. The acceleration vector of the body remain constant. The minimum radius of curvature of trajectory of the body is.
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