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### 2-Exercise-Circular Motion

• 2. 17. Fig. (b) is a graph of the angular position of the rotating disc of fig. (a). Is the angular velocity of the disc positive, negative, or zero at (a) t = 1s, (b) t = 2s, and (c) t = 3s ? (d) Is the angular acceleration positive or negative ? Type (III) : Long Answer Type Questions: [04 Mark Each] 18. Establish relation between linear and angular velocities. 19. Define angular velocity and angular acceleration. Derive an expression for centripetal acceleration in case of a uniform circular motion of an object. 20. Discuss the motion of a car on a banked circular track. 21. A 70 kg man stands in contact against the inner wall of a hollow cylindrical drum of radius 3 m rotating about its vertical axis. The co-efficient of friction between the wall and his clothing is 0.15. What is the minimum rotational speed of the cylinder to enable the man to remain stuck to the wall (without falling) when the floor is suddenly removed ?
• 6. A 7. The second’s hand of a watch has length 6 cm. Speed of end point and magnitude of difference of velocities at two perpendicular positions will be : (A) 2 & 0 mm/s (B)  2 2 & 4.44 mm/s (C)  2 2 & 2 mm/s (D) 2 &  2 2 mm/s SECTION (B) : RADIAL AND TANGENTIAL ACCELERATION B 1. Two particles P and Q are located at distances rP and rQ respectively from the axis of a rotating disc such that rP > rQ : (A) Both P and Q have the same acceleration (B) Both P and Q do not have any acceleration (C) P has greater acceleration than Q (D) Q has greater acceleration than P B 2. Let ar and at represent radial and tangential acceleration. The motion of a particle may be circular if : (A) ar = 0, at = 0 (B) ar = 0, at  0 (C) ar  0, at = 0 (D) none of these B 3. A particle is going with constant speed along a uniform helical and spiral path separately as shown in figure (in case (a), verticle acceleration of particle is negligible) (b) (A) The velocity of the particle is constant in both cases (B) The magnitude of acceleration of the particle is constant in both cases (C) The magnitude of accleration is constant in (a) and decreasing in (b) (D) The magnitude of accleration is decreasing continuously in both the cases B 4. If the radii of circular paths of two particles of same masses are in the ratio of 1 : 2, then in order to have same centripetal force, their speeds should be in the ratio of : (A) 1 : 4 (B) 4 : 1 (C) 1 : 2 (D) 2 : 1 SECTION (C) : CIRCULAR MOTION IN HORIZONTAL PLANE C 1. A stone of mass of 16 kg is attached to a string 144 m long and is whirled in a horizontal smooth surface. The maximum tension the string can withstand is 16 N. The maximum speed of revolution of the stone without breaking it, will be : (A) 20 ms–1 (B) 16 ms–1 (C) 14 ms–1 (D) 12 ms–1 C 2. On horizontal smooth surface a mass of 2 kg is whirled in a horizontal circle by means of a string at an initial angular speed of 5 revolutions per minute. Keeping the radius constant the tension in the string is doubled. The new angular speed is nearly: (A) 14 rpm (B) 10 rpm (C) 2.25 rpm (D) 7 rpm C 3. A particle is kept fixed on a uniformly rotating turn-table As seen from the ground , the particle goes in a circle, its speed is 10 cm/s and acceleration is 10 cm/s2 .The particle is now shifted to a new position to make the radius half of the original value.The new values of the speed and acceleration will be (A) 20 cm/s, 20 cm/s2 (B) 5 cm/s, 5 cm/s2 (C) 40 cm/s, 10 cm/s2 (D) 40 cm/s,40 cm/s2 C 4. A coin placed on a rotating turntable just slips if it is placed at a distance of 16 cm from the centre. If the angular velocity of the turntable is doubled, it will just slip at a distance of (A) 1 cm (B) 2 cm (C) 4 cm (D) 8 cm C 5. A rod of length L is hinged at one end and it is rotated with a constant angular velocity in a horizontal plane . Let T1 and T2 be the tensions at the points L/4 and 3L/4 away from the hinged end. (A) T1 > T2 (B) T2 > T1 (C) T1 = T2 (D) The relation between T1 and T2 depends on whether the rod rotates clockwise or anticlockwise
• 7. C 6. A particle moving along a circular path due to a centripetal force having constant magnitude is an example of motion with : (A) constant speed and velocity (B) variable speed and velocity (C) variable speed and constant velocity (D) constant speed and variable velocity. SECTION (D) : RADIUS OF CURVATURE D 1. A stone is projected with speed u and angle of projection is . Find radius of curvature at t = 0. (A) g cos u 2 2  (B)  sin g u2 (C)  cos g u2 (D) g sin u 2 2  D 2. Aparticle of mass m is moving with constant velocity v  on smooth horizontal surface. A constant force F  starts s acting on particle perpendicular to velocity v. Radius of curvature after force F start acting is : (A) F mv2 (B)  cos F mv2 (C)  sin F mv2 (D) none of these SECTION (E) : CIRCULAR MOTION IN VERTICAL PLANE E 1. A car is going on an overbridge of radius R, maintaining a constant speed. As the car is descending on the overbridge from point B to C, the normal force on it : (A) increase (B) decreases (C) remains constant (D) first increases then decreases. E 2. In a circus, stuntman rides a motorbike in a circular track of radius R in the vertical plane. The minimum speed at highest point of track will be : (A) gR 2 (B) 2gR (C) gR 3 (D) gR E 3. A particle is moving in a vertical circle. The tensions in the string when passing through two positions at angles 30° and 60° from downward vertical are T1 and T2 respectively. Then (A) T1 = T2 (B) T2 > T1 (C) T1 > T2 (D) Tension in the string always remains the same E 4. A stone tied to a string is rotated in a vertical plane. If mass of the stone is m, the length of the string is r and the linear speed of the stone is v when the stone is at its lowest point, then the tension in the string at the lowest point will be : (A) mg r mv2  (B) mg r mv2  (C) r mv2 (D) mg E 5.* A car is moving with constant speed on a road as shown in figure. The normal reaction by the road on the car is NA , NB and NC when it is at the points A, B and C respectively. (A) NA = NB (B) NA > NB (C) NA < NB (D) NC > NA
• 8. E 6. A bucket is whirled in a vertical circle with a string attached to it.The water in bucket does not fall down even when the bucket is inverted at the top of its path. In this position choose most appropiate option if v is the speed at the top. (A) mg = r mv2 (B) mg is greater than r mv2 (C) mg is not greater than r mv2 (D) mg is not less than r mv2 E-7._ A particle is moving in the vertical plane. It is attached at one end of a string of length l whose other end is fixed. The velocity at the lowest point is u. The tension in the string isT  and acceleration of the particle is  a at any position. Then, T  .  a is zero at the highest point: (A) only if u   g 2 (B) if u   g 5 (C) only if u =  g 2 (D) only if u >  g 2 E-8._ In the above question,T  .  a is non-negative at the lowest point for: (A) u   g 2 (B) u =  g 2 (C) u <  g 2 (D) any value of u E 9_ In the above question,  T . u is zero for: (A) u  2 g (B) u = 2 g (C) u  2 g (D) any value of u SECTION (F) : MOTION OF A VEHICLE, CENTRIFUGAL FORCE AND ROTATION OF EARTH F 1. A car moving on a horizontal road may be thrown out of the road in taking a turn : (A) By the gravitational force (B) Due to lack of sufficient centripetal force (C) Due to friction between road and the tyre (D) Due to reaction of earth F 2*. A curved section of a road is banked for a speed v. If there is no friction between road and tyres of the car, then: (A) car is more likely to slip at speeds higher than v than speeds lower than v (B) car cannot remain in static equilibrium on the curved section (C) car will not slip when moving with speed v (D) none of the above F 3. AtrainA runs from east to west and another train B of the same mass runs from west to east at the same speed with respect to earth along the equator. Normal force by the track on train A is N1 and that on train B is N2 : (A) N1 > N2 (B) N1 < N2 (C) N1 = N2 (D) the information is insufficient to find the relation between N1 and N2 . F-4.* A car of mass m attempts to go on the circular road of radius r, which is banked for a speed of 36 km/hr. The friction coefficient between the tyre and the road is negligible. (A) The car cannot make a turn without skidding. (B) If the car turns at a speed less than 36 km/hr, it will slip down (C) If the car turns at the constant speed of 36 km/hr, the force by the road on the car is equal to r mv2 (D) If the car turns at the correct speed of 36 km/hr, the force by the road on the car is greater than mg as well as greater than r mv2
• 10. 4. STATEMENT-1 : When a particle is projected at some angle ( 90º ) with horizontal, the radius of curvature of its path during the ascent decreases continuously. STATEMENT-2 : The radius of curvature is the ratio of square of magnitude of the velocity and the acceleration at that point. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 isTrue; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 isTrue, Statement-2 is False (D) Statement-1 is False, Statement-2 isTrue PART - I : SUBJECTIVE QUESTIONS 1. A particle is moving along a circular path of radius 5 m with a uniform speed 5 ms–1 . What is the magnitude of average acceleration during the interval in which particle completes half revolution? 2. A particle moves clockwise in a circle of radius 1 m with centre at (x, y) = (1m, 0). It starts at rest at the origin at time t = 0. Its speed increases at the constant rate of        2 m/s2 . (a) How long does it take to travel halfway around the circle? (b) What is the speed at that time? (c) What is the net acceleration at that time ? 3. Figure shows the direction of total acceleration and velocity of a particle moving clockwise in a circle of radius 2.5 m at a given instant of time. At this instant if magnitude of net acceleration is 25 m/sec2 , find : (a) the radial acceleration, (b) the speed of the particle and 2 . 5 m v a 30º (c) its tangential acceleration 4. Two particles A and B move anticlockwise with the same speed v in a circle of radius R and are diametrically opposite to each other. At t = 0, A is imparted a tangential acceleration of constant magnitude at = R 25 v 72 2  . Calculate the time in which A collides with B, the angle traced by A during this time , its angular velocity and radial acceleration at the time of collision. 5. A block of mass ‘m’ moves on a horizontal circle against the wall of a cylindrical room of radius R. The floor of the room on which the block moves is smooth but the friction coefficient between the wall and the block is µ. The block is given an initial speed v0 . As a function of the instantaneous speed ‘v’ write (i) the normal force by the wall on the block, (ii) the frictional force by the wall and (iii) the tangential acceleration of the block. (iv) obtain the speed of the block after one revolution.
• 11. 6. A uniform metallic chain in a form of circular loop of mass m with a length  rotates at the rate of n revolutions per second. Find the tension T in the chain. 7. A 4 kg block is attached to a vertical rod by means of two strings of equal length. When the system rotates about the axis of the rod, the strings are extended as shown in figure. A 8m  5m (a) How many revolutions per minute must the system make in order for the tension in the upper chord to be 20 kgf? (b) What is the tension in the lower chord ? 8. A block of mass m is kept on a horizontal ruler. The friction coefficient between the ruler and the block is . The ruler is fixed at one end and the block is at a distance L from the fixed end. The ruler is rotated about the fixed end in the horizontal plane through the fixed end. (A) What can the maximum angular speed before which the block does not slip ? (B) If the angular speed of the ruler is uniformly increased from zero at a constant angular acceleration , at what angular speed will the block slip? 9. Two identical rings which can slide along the rod are kept near the mid point of a smooth rod of length 2. The rod is rotated with constant angular velocity  about vertical axis passing through its centre. The rod is at height ' h ' from the ground. Find the distance between the points on the ground where the rings will fall after leaving the rods. 10. A small body of mass m is allowed to slide on an inclined frictionless track from rest position as shown in the figure. (i) Find the minimum height h, so that body may successfully complete the loop of radius ‘r’. (ii) If h is double of that minimum height, find the resultant force on the block at position H 11. A nail is located at a certain distance vertically below the point of suspension of a simple pendulum. The pendulum bob is released from the position where the string makes an angle of 60° from the vertical. Calculate the distance of the nail from the point of suspension such that the bob will just perform revolution with the nail as centre. Assume the length of pendulum to be 1m.
• 13. PART - II : OBJECTIVE QUESTIONS Single choice type 1. Three point particles P, Q, R move in a circle of radius ' r ' with different but constant speeds. They start moving at t = 0 from their initial positions as shown in the figure. The angular velocities (in rad/sec) of P, Q and R are 5 , 2  & 3  respectively, in the same sense. The time interval after which they all meet is: (A) 2/3 sec (B) 1/6 sec (C) 1/2 sec (D) 3/2 sec 2. The velocity and acceleration vectors of a particle undergoing circular motion are v  = î 2 m/s and a  = î 2 + ĵ 4 m/s2 respectively at an instant of time. The radius of the circle is (A) 1m (B) 2m (C) 3m (D) 4m 3. Aparticle of mass m begins to slide down a fixed smooth sphere from the top with negligible intial velocity. What is its tangential acceleration when it breaks off the sphere ? (A) 3 g 2 (B) 3 g 5 (C) g (D) 3 g 4. A boy whirls a stone in a horizontal circle 1.8 m above the ground by means of a string with radius 1.2 m. while whirling the stone string was horizontal, it breaks and stone flies off horizontally, striking the ground 9.1 m away. The centripetal acceleration during the circular motion was nearly: (use g = 9.8 m/s2 ) (A) 94 m/s2 (B) 141 m/s2 (C) 188 m/s2 (D) 282 m/s2 5. A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration ac is varying with time t as ac = k2 rt2 where k is a constant. The power delivered to the particle by the force acting on it is- (A) 2  mk2 r2 (B) mk2 r2 t (C) 3 ) t r mk ( 5 2 4 (D) Zero 6. A large mass M hangs stationary at the end of a light string that passes through a smooth fixed ring to a small mass m that moves around in a horizontal circular path. If  is the length of the string from m to the top end of the tube and  is angle between this part and vertical part of the string as shown in the figure, then time taken by m to complete one circle is equal to M m  (A)   sin g 2  (B)   cos g 2  (C)   sin M g m 2  (D) M g m 2   7. A stone of mass 1 kg tied to a light inextensible string of length L = 3 10 m, whirling in a circular path in a vertical plane. The ratio of maximum tension in the string to the minimum tension in the string is 4, If g is taken to be 10 m/s2 , the speed of the stone at the highest point of the circle is : (A) 10 m/s (B) 5 2 m/s (C) 10 3 m/s (D) 20 m/s 8. Three identical particles are joined together by a thread as shown in figure. All the three particles are moving on a smooth horizontal plane about point O. If the speed of the outermost particle is v0 , then the ratio of tensions in the three sections of the string is : (Assume that the string remains straight) (A) 3 : 5 : 7 (B) 3 : 4 : 5 (C) 7 : 11 : 6 (D) 3 : 5 : 6
• 15. 15. The kinetic energy k of a particle moving along a circle of radius R depends on the distance covered s as k = as2 where a is a positive constant. The total force acting on the particle is : (A) 2a R s2 (B) 2as 2 / 1 2 2 R s 1          (C) 2 as (D) 2a s R2 16. Consider a case in which, the driver of a car travelling at a high speed suddenly sees a wall at a distance r directly in front of him. To avoid collision, (A) he should apply brakes sharply (B) he should turn the car sharply (C) he should apply brakes and then sharply turn (D) None of these 17. A particle moves with deceleration along the circle of radius R so that at any moment of time its tangential and normal accelerations are equal in moduli. At the initial moment t = 0 the speed of the particle equals v0 , then : (i) the speed of the particle as a function of the distance covered s will be (A) v = v0 e–s/ R (B) v = v0 es/ R (C) v = v0 e–R/s (D) v = v0 eR/s (ii) the total acceleration of the particle as function of velocity and distance covered (A) a = R v 2 2 (B) a = R v2 (C) a = R v 2 2 (D) a = R v 2 2 2 18. A particle moves along an arc of a circle of radius R. Its velocity depends on the distance covered s as v = a s , where a is a constant then the angle  between the vector of the total acceleration and the vector of velocity as a function of s will be (A) tan  = s 2 R (B) tan = 2s / R (C) tan  = s R 2 (D) tan  = R 2 s 19. A particle A moves along a circle of radius R = 50 cm so that its radius vector r relative to the fixed point O (Fig.) rotates with the constant angular velocity  = 0.40 rad/s. Then modulus v of the velocity of the particle, and the modu lus a of its total acceleration will be (A) v = 0.4 m/s, a = 0.4 m/s2 (B) v = 0.32 m/s, a = 0.32 m/s2 (C) v = 0.32 m/s, a = 0.4 m/s2 (D) v = 0.4 m/s, a = 0.32 m/s2 20. A sphere of mass m is suspended by a thread of length ‘’ is oscillating in a vertical plane, the angular amplitude being 0 . What is the tension in the thread when it makes an angle  with the vertical during oscillations ? If the thread can support a maximum tension of 2 mg, then what can be the maximum angular amplitude of oscillation of the sphere without breaking the rope? (A) 3 mg cos – 2mg cos0 0 = 600 (B) 3 mg cos + 2mg cos0 ,, 0 = 600 (C) 2mg cos – 3mg cos0 0 = 300 (D) 2mg cos + 3mg cos0 ,, 0 = 300 21. Two small spheres, each of mass m are rigidly connected by a rod of negligible mass and are released from rest in the position shown and slide down a smooth circular guide in the vertical plane. (Using information that massless rod does not exert force on spheres in the direction normal to the rod if contact is at only two points) m 45° r r r 2 1 2 y x 1 A m m m