One end of a light string of length L is connected to a ball and the other end is connected to a fixed point O. The ball is released from rest at t = 0 with string horizontal and just taut. The ball then moves in vertical circular path as shown. The time taken by ball to go from position A to B is t_{1} and from B to lowest position C is t_{2}. Let the velocity of ball at B is and at C is respectively. If then, which of the following is true.
Tangential acceleration is a_{t} = g cos θ which decrease with time.
Hence the plot of a_{t} versus time may be as shown in graph.
Area under graph in time interval t_{1} = v_{B} – 0 = v_{B}
Area under graph in the time interval t_{2} = v_{C} – v_{B} = v_{B}
Hence the under graph in time t_{1} and t_{2} is same.
∴ t_{1} < t_{2}
∴ t_{1} < t_{2}
The correct answers are:
A machine, in an amusement park, consists of a cage at the end of one arm, hinged at O. The cage revolves along a vertical circle of radius r (ABCDEFGH) about its hinge O, at constant linear speed The cage is so attached that the man of weight ‘w’ standing on a weighing machine, inside the cage, is always vertical. Then which of the following is/are correct.
Let N be the normal reaction (Reading of the weighing machine)
Put v ∴ NA – mg = mg
⇒ N_{A} = 2mg = 2W
hence N_{A} > N_{E} by 2W
Now at G, N_{G} = mg = W = N_{c}
The correct answers are: the weight reading at A is greater than the weight reading at E by 2 w, the weight reading at G = w, the ratio of the weight reading at E to that at A = 0, the ratio of the weight reading at A to that at C = 2
A particle is moving in a circular path. The acceleration and momentum vectors at an instant of time are Then which of the following statement is incorrect
A car driver going at some speed suddenly finds a wide wall at a distance r. To avoid hitting the wall he should.
When he applies brakes
if µ is the friction coefficient then a = µg
when he takes turn
Then we can see r < s_{1} hence driver can hit the wall when he take turn due to insufficient radius of curvature.
The correct answer is: apply the brakes
Two particles A and B separated by a distance 2R are moving counter clockwise along the same circular path of radius R each with uniform speed v. At time t = 0. A is given a tangential acceleration of magnitude
As when they collide
Now angle covered by
Put t ∴ angle covered by
The correct answers are: the time lapse for the two bodies to collide is the angle covered of A is 11π/6
A smooth rod PQ rotates in a horizontal plane about its mid point M which is h = 0.1 m vertically below a fixed point A at a constant angular velocity 14 rad/s. A light elastic string of natural length 0.1 m force constant 1.47 N/cm has one end fixed at A and its other end attached to a ring of mass m = 0.3 kg which is free to slide along the rod. When the ring is stationary relative to rod, then which of following is correct.
but R = K_{x}
(K = 1.47 × 10^{2} N/m)
Also r = 0.1 tan θ
put T, r, m and ω in equation (2)
We have cos θ = 3/5 and T = 9.8 N
The correct answers are: Tension in spring is 9.8 N, Inclination of string with vertical is cos^{–1}(3/5)
On a circular table, A and B are moving on the circumference. Man A runs behind man B to catch him A runs with constant angular speed ω_{1} with respect to table and B runs at constant tangential speed v_{2} with respect to ground. If it is found that the table rotates 30° in the opposite direction in every one second and the initial angular separation between A and B is 30°, then A catches B after : (Radius of table is 3 m).
The correct answers are: 0.5 sec, if A can not catch B within 0.5 sec
A small sphere of mass m suspended by a thread is first taken aside so that the thread forms the right angle with the vertical and then released, then :
By putting the value of V_{B}
T_{B} = 3mg cosθ
When total acceleration vector directed horizontally
The correct answers are: total acceleration of sphere as a function of thread tension as a function of θ is T = 3mg cos θ between the thread and the vertical at the moment when the total acceleration vector of the sphere is directed horizontal is
A section of fixed smooth circular track of radius 20 m, in vertical plane is shown in the figure. A block is released from position A and leaves the track at B :
By energy conservation between A and B
Now, radius of curvature
The correct answers are: velocity at point B is Radius of curvature of trajectory when leaves point B is R/2
A particle is describing circular motion in a horizontal plane in contact with the smooth inside surface of a fixed right circular cone with its axis vertical and vertex down. The height of the plane of motion above the vertex is h and the semivertical angle of the cone is α. The period of revolution of the particle :
Thus when h increases T also increases.
The correct answers are: increases as h increases, increases as α increases
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