1) For a block of mass m to slide without friction up a rise of height h, the minimum initial speed of the block must be

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1 v m 1) For a bock of mass m to side without friction up a rise of height h, the minimum initia speed of the bock must be a ) gh b ) gh d ) gh e ) gh c ) gh P h b 3 15 ft 3) A man pus a pound crate up a frictioness 3 sope 15 feet high. If the crate is moved from rest at the bottom to rest at the top of the sope, the work done by the man is a) +3 ft-b b) +15 ft-b c) d) -15 ft-b e) -3 ft-b Vo h max θ 5 m/s ) A projectie has an initia veocity of vo which makes an ange with the horizonta. Negecting air friction, a of the foowing are true at point P, the highest point of its trajectory, EXCEPT : a) The time t required to reach P is given by t = v cosθ g b) The horizonta dispacement has been votcosθ where t is the time required to reach P. c) hmax = (v osin θ ) g d) The speed is vocosθ e) The acceeration is g. θ 1m 4) An object sides off a roof 1 m above the ground with an initia horizonta speed of 5 m/s as shown above. The time between the object's eaving the roof and hitting the ground is most neary a) 1 e)5 s s b) 1 s c) s d) s Refer to this graph for questions 5 and 6

2 v car y v car x t(s) At time t=, car x traveing with speed vo passes car y which is just starting to move. Both cars then trave on two parae anes of the same straight road. The graphs of speed v versus time t for both cars are shown above. 5) Which of the foowing is true at t= seconds? a) Car y is behind car x. b) Car y is passing car x. c) Car y is in front of car x. d) Both cars have the same acceeration. e) Car x is acceerating faster than car y. 6) From t= to t=4 seconds, the areas under both curves are equa. Therefore which of the foowing is true at t=4 seconds? a) Car y is behind car x. b) Car y is passing car x. c) Car y is in front of car x. d) Both cars have the same acceeration. e) Car x is acceerating faster than car y. m 1 m A system consists of two objects having masses m 1 and m (m 1 <m ). The objects are connected by a massess string, hung over a puey as shown above, and then reeased. 7) When the speed of each object is v, the magnitude of the tota inear momentum of the system is a) ( m 1 +m )v b) ( m 1 - m )v ( ) m c) 1 +m e) m v v d ) ( ) m 1 - m v 8) When the object of mass m has descended a distance h, the potentia energy of the system has decreased by ( ) gh b ) m 1gh a ) m - m 1 Refer to the system beow for questions 7 and 8 c ) ( m 1 + m )gh d ) 1 ( m 1 + m )gh e) 9) The foowing graphs, a drawn to the same scae, represent the net force F as a function of dispacement x for an object that moves aong a straight ine. Which graph

3 represents the force that wi cause the greatest change in the kinetic energy of the object from x= to x=x 1? Partice C v(m/s) +1. a) b) F F +1.. t(s) -1. x1 x c) F d) F x1 x Partice D v(m/s) +1. e) F x1 x x1 x t(s) x1 x Partice E v(m/s) +1. Questions 1,11 and 1 reate to five partices that start at x= at t= and move in one dimension independenty of one another. Graphs of the veocity of each partice versus time are shown beow t(s) Partice A v(m/s) Partice B v(m/s) t(s) -1. t(s) 1) Which partice is farthest from the origin at seconds? a) A b) B c) C d) D e) E 11) Which partice moves with a constant non zero acceeration? a) A b) B c) C d) D e) E 1) Which partice is in its initia position at t= seconds? a) A b) B c) C d) D e) E 13) A ba of mass m at one end of a string of ength R rotates in a vertica circe just fast enough to keep the string from going sack when the ba is at the top of the circe. The ba s speed when it is at the bottom of the circe is

4 Vx Vx a) gr b) 3gR c ) 4gR a) b) d) 5gR e) 7gR t t +y Vx c) d) Vx t t +x e) Vx t A bock sides down a frictioness incine as indicated in the figure above. Let y=, x=, and t= just as the bock is reeased from rest at the top of the incine. Let the positive direction for y be up and the positive direction for x be to the right. 14) Which of the foowing graphs best represents the x component of the veocity of the bock as a function of time? m, v m, v 6 6 y x 15) Two partices of equa mass m o, moving with equa speeds v o, aong paths incined at 6 degrees to the x axis as shown above, coide and stick together. Their veocity after the coision has magnitude a) v 4 b) v c) v d) 3 v e) v

5 O 3 18) An unstabe nuceus has mass M and is initiay at rest. It ejects a partice of mass m with speed v o. The remaining nuceus recois in the opposite direction with a speed 16)A uniform rigid bar of weight W is supported in a horizonta orientation as shown above by a rope that makes a 3 degree ange with the horizonta. The force exerted on the bar at point O, where it is pivoted, is best represented by a vector whose direction is which of the foowing? a) v b) m M v c) ( m M+m )v d) ( m +M m )v m e) ( M- m )v 19) The Newton-second is a unit of: a) b) c) a)work c)power e)none of these b) anguar momentum d) inear momentum d) e) ) A 6-kg partice moves to the right at 4 m/s as shown. Its anguar momentum in kg -m/s about the point O is: M 1 v M 17) Mass M1 is moving toward stationary mass M. The speed of the center of mass of the system is a) zero b)88 c) 144 d) 4 e)49 a) ( M 1 M )v b) (1+ M 1 M )v c) (1+ M M 1 )v M 1 d) (1- M 1 )v e) ( )v M M 1 +M 1) Two disks are mounted on ow friction bearings on a common shaft. The first disc has rotationa inertia I and is spinning with anguar veocity ω. The second disc has rotationa inertia I and is spinning (in the

6 same direction as the first disc) with anguar veocity ω as shown. The two disks are sowy forced toward each other aong the shaft unti they coupe and have a fina common anguar veocity of: a) 1 5 d) v g 7 v 1 g b) v 5 g e) v g c) 1 v g 4) A partice is moving in a circe of radius meters according to the reation θ = 3t + t, where θ is measured in radians and t in seconds. The speed of the partice at t = 4 seconds is a) 13 m/s b) 16 m/s c) 6 m/s d) 5 m/s e) 338 m/s ) A figure skater is spinning on frictioness ice with her arms fuy extended horizontay. She then drops her arms to her sides. Which of the foowing correcty describes her rotationa kinetic energy and anguar momentum as her arms fa? Rotationa Kinetic Anguar Energy Momentum a) Remains constant Remains constant b) Decreases Increases c) Decreases Decreases d) Increases Decreases e) Increases Remains constant 3) A bowing ba of mass M and radius R, whose moment of inertia about its center is /5 MR, ros without sipping aong a eve surface at speed v. The maximum height to which it can ro if it ascends an incine is 5) If a partice moves in such a way that its position x is described as a function of time t by x = t 3, then its kinetic energy is proportiona to a) b) t c) t d) e) 3 t 1 t t (i.e. kinetic energy is constant) 6) A partice of mass m moves aong a straight path with a speed v defined by the function v = bt + c, where b and c are constants and t is time. What is the magnitude F of the net force on the partice at time t = t 1? a) bt 1 + c b) 3mbt 1 + c c) mbt 1 d) mbt 1 + c e) mbt 1

7 I M 7) A 1 kg. body is constrained to move aong the x-axis. The potentia energy U of the body in joues is given as a function of its position x in meters by U(x) = 6x - 4x + 3. The force on the partice at x = 3 meters is a) 3 N in the +x direction b) 3 N in the -x direction c) 45 N in the +x direction d) 45 N in the -x direction e) 98 N in the +x direction 3m pivot point 8) A ight rod with masses 3m and m, attached to its ends is pivoted about a horizonta axis as shown above. When reeased from rest in a horizonta orientation, the rod begins to rotate with an anguar acceeration of magnitude m 9) A uniform stick has ength. The moment of inertia about the center of the stick is I. A partice of mass M is attached to one end of the stick. The moment of inertia of the combined system about the center of the stick is a) I + 1 M 4 b) I + 1 M c) I + 3 M 4 d) I + M 5 e) I + M 4 3) At a particuar instant, a stationary observer on the ground sees a package faing with speed v 1 at an ange to the vertica. To a piot fying horizontay at constant speed reative to the ground, the package appears to be faing verticay with a speed v at that instant. What is the speed of the piot reative to the ground? a)v 1 + v b) v 1 - v c) v + v 1 d) v 1 v e) v 1 + v a) 1 g 7 d) 5 g 7 b) 1 5 g e) g c) 1 4 g v f v i 31)A ba initiay moves horizontay with

8 veocity v i, as shown above. It is then struck by a stick. After eaving the stick, the ba moves verticay with a veocity v f, which is smaer in magnitude than v i. Which of the foowing vectors best represents the direction of the average force that the stick exerts on the ba?.3 kg a) b).6 kg c) d) e).6 kg 3) A ba is thrown upward. At a height of 1 meters above the ground, the ba has a potentia energy of 5 joues (where U= at ground eve) and is moving upward with a kinetic energy of 5 joues. Air friction is negigibe. The maximum height reached by the ba is most neary a) 1 m b) m c) 3 m d) 4 m e) 5 m 33) Two.6 kiogram objects are connected by a thread that passes over a ight, frictioness puey, as shown above. The objects are initiay hed at rest. If a third object with a mass of.3 kiogram is added on top of one of the.6 kiogram objects as shown and the objects are reeased, the magnitude of the acceeration of the.3 kiogram object is most neary a) 1. m/s b) 6. m/s c) 3. m/s d). m/s e) 1. m/s 34) During a certain time interva, a constant force deivers an average power of 4 watts to an object. If the object has an average speed of m/s and the force acts in the direction of motion of the object, the magnitude of the force is a) 16 N b) 8 N c) 6N d) 4 N e) N

9 X 1m/s X 6m/s 53 a) v= figure I Y figure II b) 1 m/s 35) Two bas of equa mass are on a frictioness horizonta tabetop. Ba X initiay moves at 1 m/s, as shown in figure I above. It then coides easticay with identica ba Y, which is initiay at rest. After the coision, ba X moves at 6 m/s aong a path at 53 to its origina direction, as shown in figure II above. Which of the foowing diagrams best represents the motion of ba Y after the coision? c) d) m/s 8 m/s e) 53 8 m/s probems 36 & 37 refer to this figure An ant of mass m cings to the rim of a

10 fywhee of radius r, as shown above. The fywhee rotates cockwise on a horizonta shaft S with constant anguar veocity w. As the whee rotates, the ant revoves past the stationary points I, II, III, and IV. The ant can adhere to the whee with a force much greater than its own weight. 36) It wi be most difficut for the ant to adhere to the whee as it revoves past which of the four points? a) I b) II c) III d) IV e) It wi be equay difficut for the ant to adhere to the whee at a points. 37) What is the magnitude of the minimum adhesion force necessary for the ant to stay on the fywhee at point III? a) mg b) mwr c) mwr + mg d) mwr - mg e) mwr + mg 38)A weight ifter ifts a mass m at constant speed to a height h in time t. How much work is done by the weight ifter? a) mg b) mh c)mgh d) mght e) mgh/t 39)A conservative force has the potentia energy function U(x), shown by the graph above. A partice moving in one dimension under the infuence of this force has kinetic energy 1. joue when it is at position x1. Which of the foowing is a correct statement about the motion of the partice? a) It osciates with maximum position x and minimum position x. b) It moves to the right of x3 and does not return. c) It moves to the eft of x and does not return. d) It comes to rest at either x or x. e) It cannot reach either x or x. 4) An object is shot verticay upward into the air with a positive initia veocity. (i.e. upward is the positive direction) Which of the foowing correcty describes the veocity and acceeration of the object at its maximum eevation? Veocity Acceeration a) Positive Positive b) Zero Zero

11 c) Negative Negative d) Zero Negative e) Positive Negative diagram for probem 43 41) A turntabe that is initiay at rest is set in motion with a constant anguar acceeration α. What is the anguar veocity of the turntabe after it has made one compete revoution? a ) α b ) πα c ) 4 πα d ) α e) 4πα v 43) Two identica massess springs are hung from a horizonta support. A bock of mass 1. kiograms is suspended from the pair of springs, as shown above. When the bock is in equiibrium, each spring is stretched an additiona.15 meter. m The force constant of each spring is most neary a) 4 N/m b) 48 N/m c) 6 N/m 4) An object of mass m is moving with speed v to the right on a horizonta frictioness surface, as shown above, when it expodes into two pieces. Subsequenty, one piece of mass 5 m moves with a speed v to the eft. The speed of the other piece d) 8 N/m e) 96 N/m of the object is a) v b) v 3 c) 7v 5 d) 3v e) v 44) The rigid body shown in the diagram above consists of a vertica support post

12 and two horizonta crossbars with spheres attached. The masses of the spheres and the engths of the crossbars are indicated in the diagram. The body rotates about a vertica axis aong the support post with constant anguar speed w. If the masses of the support posts and the crossbars are negigibe, what is the ratio of the anguar momentum of the two upper spheres to that of the two ower spheres? a) /1 b) 1/1 c) 1/ d) 1/4 e) 1/8 46) Which of the five ettered points represents the center of mass of the sphere-rod combination? a)a b)b c)c d)d e)e 47) The rod-sphere combination can be pivoted about an axis that is perpendicuar to the pane of the page and that passes through one of the five ettered points. Through which point shoud the axis pass for the moment of inertia of the sphere-rod combination about this axis to be greatest? 45) A spring oaded gun can fire a projectie to a height h if it is fired straight up. If the same gun is pointed at an ange of 45 from the vertica, what maximum height can now be reached by the projectie? a)a b)b c)c d)d e)e a ) h 4 d ) h b ) h e )h c) h F 5 kg 3m this figure refers to probems 46 & 47 A 1kg B C D E 5kg A 5 kiogram sphere is connected to a 1 kiogram sphere by a rigid rod of negigibe mass, as shown above. 4m 48) A bock of mass 5 kiograms ies on an incined pane, as shown above. The horizonta and vertica supports for the pane have engths of 4 meters and 3 meters respectivey. The coefficient of friction between the pane and the bock is.3. The magnitude of the force F necessary to pu the bock up the pane with constant speed is most neary

13 a) 3 N b) 4 N a)m 1 + M b) M 1 + M c)m 1 M c) 49 N d) 5 N e) 58 N d) M 1 M e) M 1 M 1 5) In the absence of air friction, an object dropped near the surface of the Earth experiences a constant acceeration of about 9.8 m/s. This means that the m M 1 a) speed of the object increases 9.8 m/s during each second b) speed of the object as it fas is is 9.8 m/s 1 c) object fas 9.8 meters during each second M m d) object fas 9.8 meters during the first second ony e) derivative of the distance with respect to time for the object equas 9.8 m/s. 51) A 5 kg sports car acceerates uniformy from rest, reaching a speed of 3 m/s in 6 seconds. During the 6 seconds, the car has traveed a distance of 49)A rod of negigibe mass is pivoted at a point that is off center, so that ength 1 is different from ength. The figures above show two cases in which masses are suspended from the ends of the rod. In each case the unknown mass m is baanced by a known mass, M1 or M so that the rod remains horizonta. What is the vaue of m in terms of the known masses? a) 15 m b) 3 m c) 6 m d) 9 m e) 18 m

14 5) If F 1 is the magnnitude of the force exerted by the Earth on a sateite in orbit about the Earth and F is the magnitude of the force exerted by the sateite on the Earth, then which of the foowing is true? a) F 1 is much greater than F. b) F 1 is sighty greater than F. c) F 1 is equa to F. d) F is sighty greater than F 1. e) F is much greater than F 1. questions x= -A x= x= A A bock on a horizonta frictioness pane is attached to a spring, as shown above. The bock osciates aong the x- axis with simpe harmonic motion of ampitude A. 53) Which of the foowing statements about the bock is correct? a) At x=, its veocity is zero. b) At x=, its acceeration is at a maximum. c) At x=a, its dispacement is at a maximum. d) At x=a, its veocity is at a maximum. e) At x=a, its acceeration is zero. 54) Which of the foowing statements about energy is correct? a) The potentia energy of the spring is at a minimum at x=. b) The potentia energy of the spring is at a minimum at x=a. c) The kinetic energy of the bock is at a minimum at x=. d) The kinetic energy of the bock is at a maximum at x=a. e) The kinetic energy of the bock is aways equa to the potentia energy of the spring. 55) A baoon of mass M is foating motioness in the air. A person of mass ess than M is on a rope adder hanging from the baoon. The person begins to cimb the adder at a uniform speed v reative to the ground. How does the baoon move reative to the ground? a) Up with speed v. b) Up with a speed ess than v. c) Down with speed v. d) Down with a speed ess than v. e) The baoon does not move. 56) If one knows ony the constant resutant force acting on an object and the time during which this force acts, one can determine the a) change in momentum of the object b) change in veocity of the object c) change in kinetic energy of the object d) mass of the object e) acceeration of the object

e) I, II, and III diagram for probem 57 58) A simpe penduum consists of a 1. kg brass bob on a string about 1. m ong. It has a period of. seconds. The penduum woud have a period of 1.

15 e) I, II, and III diagram for probem 57 58) A simpe penduum consists of a 1. kg brass bob on a string about 1. m ong. It has a period of. seconds. The penduum woud have a period of 1. second if the 57) Two identica massess springs are hung from a horizonta support. A bock of mass 1. kiograms is suspended from the pair of springs, as shown above. When the bock is in equiibrium, each spring is stretched an additiona.15 meter. When the bock is set into osciation with ampitude A, it passes through its equiibrium point with a speed v. In which of the foowing cases wi the bock, when osciating with ampitude A, aso have speed v when it passes through its equiibrium point? I. The bock is hung from ony one of the two springs. II. The bock is hung from the same two springs, but the springs are connected in series {on top of each other} rather than in parae {as shown}. III. A.5 kg mass is attached to the bock. a) None b) III ony c) I and II ony d) II and III ony a) string were repaced by one about.5 m ong b) string were repaced by one about. m ong c) bob were repaced by a.5 kg brass sphere d) bob were repaced by a 4. kg brass sphere e) ampitude of the motion were incteased The graph beow shows the veocity of an object moving aong a straight ine as a function of time. veocity(m/s) time (sec) 59) How far from its position at time t= was the object at t=6 seconds? a) m b) 1.5 m c).5 m d) 3.5 m e) 4.5 m

16 6) For times t<1 second the acceeration is constant. At what vaue of t was the veocity equa to zero? a) 1 sec b) sec c) -1 sec d) -1.5 sec e) - sec a) b) c) d) e) g g 3g 4g 5g 6) Two objects of equa mass on a smooth horizonta tabe are connected by strings A and B of equa ength to a fixed point O as shown beow. 61) The moment of inertia of a uniform rod of ength and mass m about an axis through one end of the rod and m perpendicuar to the rod is 3. A uniform rod 1 meter ong is pivoted at one end. It is reeased from the horizonta position and aowed to swing freey from rest. Find the transationa speed of the ower end of the rod at the instant it passes point P. Pivot A The system consisting of the two objects and the two strings rotates in a horizonta pane at a constant anguar veocity w about O. The ratio of the tension in string B to the tension in string A is a) 1:1 b) :1 c) :3 d) 1: e) 3:4 B P v

PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE)

Cass XI TARGET : JEE Main/Adv PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) ALP ADVANCED LEVEL LPROBLEMS ROTATION- Topics Covered: Rigid body, moment of inertia, parae and perpendicuar axes theorems,