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question_answer1) Consider the situation shown in figure. The system is released from rest and the block of mass 1.0 kg is found to have a speed 0.3 m/s after it has descended through a distance of 1 m. Find the coefficient of kinetic friction between the block and the table.
question_answer2) In an XY horizontal plane a force field \[\vec{F}=-\left( 40N/m \right)\left( y\hat{i}+x\hat{j} \right)\]is present where x and y the coordinates of any point on the plane. A smooth rod AB is fixed in the plane as shown in the figure. A particle of mass 5 kg is to be released with a velocity in this force field such that I reaches to point B. Find the minimum velocity (in m/s) that must be imparted along the rod at A such that it reaches to B.
question_answer3) One end of a block of mass m = 3 kg placed on a horizontal smooth surface is attached with a spring of force constant 100 N/m. Other end of the spring is attached to a wall as shown in figure. Suddenly a constant horizontal force of 10 N starts acting on the block towards right and as a result the block starts moving towards right. Find energy (in J) stored in the spring when the block stops for the first time (initially spring was in natural length).
question_answer4) A system consists of two identical cubes, each of mass 3 kg, linked together by a compressed weightless spring of force constant 1000 N/m. The cubes are also connected by a thread which is burnt at a certain moment. At what minimum value of initial compression \[{{x}_{0}}\](in cm) of the spring will the lower cube bounce up after the thread is burnt through?
question_answer5) A mass m is attached to two spring having spring constant k is in equilibrium as shown in the figure. The work done to lift the block upward at a distance mg/4k, is\[=\frac{1}{k}{{\left( \frac{mg}{n} \right)}^{2}}\]. Find n?
question_answer6) A particle of mass \[\frac{10}{7}\] Kg is moving in the positive direction of \[x\]. Its initial position \[x=0\] & initial velocity is 1 m/s. The velocity at \[x=10\] is (in m/s)
question_answer7) The potential energy of a particle is determined by the expression\[U=a\left( {{x}^{2}}+{{y}^{2}} \right)\], where \[\alpha \] is a positive constant. The particle begins to move from a point with the coordinates (3, 3) (m), only under the action of potential field force. Then its kinetic energy T at the instant when the particle is at a point with the coordinates (1, 1) (m) is 32\[\alpha /n\]. Find n.
question_answer8) A cyclist is going in a straight line at a uniform velocity 18 km/h. The resistance force can be expressed as force \[k{{v}^{2}}\]where \[k=0.8\] where velocity is in m/s and unit of force is newton. The mass of the cyclist with the bicycle is 100 kg. Neglect the rolling resistance force. If the power of the cyclist during the ride is \[{{10}^{x}}\]watt, then what is x?
question_answer9) A small block of mass 20 kg rests on a bigger block of mass 30 kg, which lies on a smooth horizontal plane. Initially the whole system is at rest. The coefficient of friction between the blocks is 0.5. The horizontal force, F = 50N, is applied on the lower block. The work done by frictional force on upper block in \[t=2s\] is \[n\times 10J\]. Find n.
question_answer10) Given\[\vec{F}=\left( x{{y}^{2}}\hat{i}+{{x}^{2}}y\hat{j} \right)N\]. Find work done by \[\vec{F}\]when a particle is taken along the semicircular path OAB where the co-ordinates of B are (2, 2) and O is origin is.
question_answer11) A small ball rests at the bottom of a watch glass of radius R. It is displaced through a small distance x from this position and released. The total distance covered before it comes to the bottom and rests there is\[\frac{{{x}^{2}}}{a\mu R}\]. (Coefficient of friction between watch glass surface and the ball is\[\mu \].) Determine a.
question_answer12) A body of mass m is hauled from the Earth's surface by applying a force F varying with the height of ascent y as \[\vec{F}=2\left( ay-1 \right)m\overset{\scriptscriptstyle\rightharpoonup}{g}\], where a is a positive constant. The work performed by this force in the gravitational field of the Earth over the first half of the ascent is \[x\frac{mg}{4a}\]. Determine x.
question_answer13) A body of mass m is thrown at an angle \[\alpha \] to the horizontal with initial velocity\[{{v}_{0}}\]. Find the mean power imparted by gravity over the whole time of motion of body and the instantaneous power of gravity as a function of time.
question_answer14) One end of a light spring of natural length d and spring constant K is fixed on a rigid wall and the other is fixed to a smooth ring of mass m which can slide without friction in a vertical rod fixed at a distance d from the wall. Initially the spring makes an angles of \[37{}^\circ \]with the horizontal as shown in figure. When the system is released from rest, find the speed (in m/s) of the ring when the spring becomes horizontal. [\[\sin 37{}^\circ =3/5,\frac{g}{d}=2\]unit and \[\frac{k}{m}=16\]unit]
question_answer15) A 1 kg block collides with a horizontal weightless spring of force constant 2.75 N/m as shown in figure. The block compresses the spring 4 m from the free position. If the coefficient of kinetic friction between the block and the horizontal surface is 0.25, find the speed (in m/s) of the block at the instant of collision in m/s. (g=10 \[m/{{s}^{2}}\])
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