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Page 1 Solved Examples on Rotational Motion JEE Mains Single Correct Q1. The velocities of three particles of masses ???? ?? , ???? ?? and ???? ?? are ???? ???, ???? ???, and ???? ?? ? ? ? respectively. The velocity of the centre of mass of the three particles is (a) ?? ??? + ?? ??? + ?? ?? ? ? ? (b) ???? ( ??? + ??? + ?? ? ? ? ) (c) ???? ??? + ???? ??? + ?? ?? ? ? ? (d) ?? ??? + ???? ??? + ???? ?? ? ? ? Ans : (a) Velocity of centre of mass ?? ???? = ?? 1 ?? 1 +?? 2 ?? 2 +?? 3 ?? 3 ?? 1 +?? 2 +?? 3 = 20×10??ˆ+30×10?? ˆ+50×10?? ˆ 100 = 2??ˆ + 3??ˆ + 5?? ˆ . Q1. Angular displacement ( ?? ) of a flywheel varies with time as ?? = ???? + ?? ?? ?? + ?? ?? ?? then angular acceleration is given by (a) ?? + ?? ???? - ?? ?? ?? ?? (b) ?? ?? - ?? ?? (c) ?? + ?? ?? - ?? ?? (d) ?? ?? + ?? ???? Ans: (d) Angular acceleration ?? = ?? 2 ?? ?? ?? 2 = ?? 2 ?? ?? 2 ( ???? + ?? ?? 2 + ?? ?? 3 ) = 2?? + 6???? Q 2. A wheel is at rest. Its angular velocity increases uniformly and becomes ?????????? /?????? after 5 sec. The total angular displacement is (a) ???????????? (b) ?????????? (c) ???????????? (d) ???????????? Ans: (d) Angular acceleration ?? = ?? 2 -?? 1 ?? = 60-0 5 = 12rad /sec 2 Now from ?? = ?? 1 ?? + 1 2 ?? ?? 2 = 0 + 1 2 ( 12) ( 5) 2 = 150rad . Q 3. If the position vector of a particle is ??? ? = ( ?? ??ˆ + ?? ??ˆ) meter and its angular velocity is ??? ? ?? = ( ??ˆ + ?? ?? ˆ ) ?????? /?????? then its linear velocity is (in ?? /?? ) (a) ( ?? ??ˆ - ?? ??ˆ + ?? ?? ˆ ) (b) ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) (c) -( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) (d) ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) Ans: (a) ?? ? ? = ?? ? ? ? × ?? ? = ( 3??ˆ + 4??ˆ + 0?? ˆ )× ( 0??ˆ + ??ˆ + 2?? ˆ ) = | ??ˆ ??ˆ ?? ˆ 3 4 0 0 1 2 | = 8??ˆ - 6??ˆ + 3?? ˆ Page 2 Solved Examples on Rotational Motion JEE Mains Single Correct Q1. The velocities of three particles of masses ???? ?? , ???? ?? and ???? ?? are ???? ???, ???? ???, and ???? ?? ? ? ? respectively. The velocity of the centre of mass of the three particles is (a) ?? ??? + ?? ??? + ?? ?? ? ? ? (b) ???? ( ??? + ??? + ?? ? ? ? ) (c) ???? ??? + ???? ??? + ?? ?? ? ? ? (d) ?? ??? + ???? ??? + ???? ?? ? ? ? Ans : (a) Velocity of centre of mass ?? ???? = ?? 1 ?? 1 +?? 2 ?? 2 +?? 3 ?? 3 ?? 1 +?? 2 +?? 3 = 20×10??ˆ+30×10?? ˆ+50×10?? ˆ 100 = 2??ˆ + 3??ˆ + 5?? ˆ . Q1. Angular displacement ( ?? ) of a flywheel varies with time as ?? = ???? + ?? ?? ?? + ?? ?? ?? then angular acceleration is given by (a) ?? + ?? ???? - ?? ?? ?? ?? (b) ?? ?? - ?? ?? (c) ?? + ?? ?? - ?? ?? (d) ?? ?? + ?? ???? Ans: (d) Angular acceleration ?? = ?? 2 ?? ?? ?? 2 = ?? 2 ?? ?? 2 ( ???? + ?? ?? 2 + ?? ?? 3 ) = 2?? + 6???? Q 2. A wheel is at rest. Its angular velocity increases uniformly and becomes ?????????? /?????? after 5 sec. The total angular displacement is (a) ???????????? (b) ?????????? (c) ???????????? (d) ???????????? Ans: (d) Angular acceleration ?? = ?? 2 -?? 1 ?? = 60-0 5 = 12rad /sec 2 Now from ?? = ?? 1 ?? + 1 2 ?? ?? 2 = 0 + 1 2 ( 12) ( 5) 2 = 150rad . Q 3. If the position vector of a particle is ??? ? = ( ?? ??ˆ + ?? ??ˆ) meter and its angular velocity is ??? ? ?? = ( ??ˆ + ?? ?? ˆ ) ?????? /?????? then its linear velocity is (in ?? /?? ) (a) ( ?? ??ˆ - ?? ??ˆ + ?? ?? ˆ ) (b) ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) (c) -( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) (d) ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) Ans: (a) ?? ? ? = ?? ? ? ? × ?? ? = ( 3??ˆ + 4??ˆ + 0?? ˆ )× ( 0??ˆ + ??ˆ + 2?? ˆ ) = | ??ˆ ??ˆ ?? ˆ 3 4 0 0 1 2 | = 8??ˆ - 6??ˆ + 3?? ˆ Q 4. A circular disc ?? of radius ?? is made from an iron plate of thickness ?? , and another disc ?? of radius ?? ?? is made from an iron plate of thickness ?? ?? . Then the relation between the moment of inertia ?? ?? and ?? ?? is (a) ?? ?? = ???? ?? ?? (b) ?? ?? = ???? ?? ?? (c) ?? ?? = ???? ?? ?? (d) ?? ?? = ?? ?? Ans: (a) Moment of Inertia of disc ?? = 1 2 ?? ?? 2 = 1 2 ( ?? ?? 2 ???? ) ?? 2 = 1 2 ?????? ?? 4 [As ?? = ?? × ?? = ?? ?? 2 ???? where ?? = thickness, ?? = density] ? ?? ?? ?? ?? = ?? ?? ?? ?? ( ?? ?? ?? ?? ) 4 ? ?? ?? ?? ?? = 1 4 ( 4) 4 = 64 ? ?? ?? = 64?? ?? [If ?? = constant] [Given ?? ?? = 4?? ?? , ?? ?? = ?? ?? 4 ] Q 5. Three rings each of mass ?? and radius ?? are arranged as shown in the figure. The moment of inertia of the system about ?? ?? ' will be Page 3 Solved Examples on Rotational Motion JEE Mains Single Correct Q1. The velocities of three particles of masses ???? ?? , ???? ?? and ???? ?? are ???? ???, ???? ???, and ???? ?? ? ? ? respectively. The velocity of the centre of mass of the three particles is (a) ?? ??? + ?? ??? + ?? ?? ? ? ? (b) ???? ( ??? + ??? + ?? ? ? ? ) (c) ???? ??? + ???? ??? + ?? ?? ? ? ? (d) ?? ??? + ???? ??? + ???? ?? ? ? ? Ans : (a) Velocity of centre of mass ?? ???? = ?? 1 ?? 1 +?? 2 ?? 2 +?? 3 ?? 3 ?? 1 +?? 2 +?? 3 = 20×10??ˆ+30×10?? ˆ+50×10?? ˆ 100 = 2??ˆ + 3??ˆ + 5?? ˆ . Q1. Angular displacement ( ?? ) of a flywheel varies with time as ?? = ???? + ?? ?? ?? + ?? ?? ?? then angular acceleration is given by (a) ?? + ?? ???? - ?? ?? ?? ?? (b) ?? ?? - ?? ?? (c) ?? + ?? ?? - ?? ?? (d) ?? ?? + ?? ???? Ans: (d) Angular acceleration ?? = ?? 2 ?? ?? ?? 2 = ?? 2 ?? ?? 2 ( ???? + ?? ?? 2 + ?? ?? 3 ) = 2?? + 6???? Q 2. A wheel is at rest. Its angular velocity increases uniformly and becomes ?????????? /?????? after 5 sec. The total angular displacement is (a) ???????????? (b) ?????????? (c) ???????????? (d) ???????????? Ans: (d) Angular acceleration ?? = ?? 2 -?? 1 ?? = 60-0 5 = 12rad /sec 2 Now from ?? = ?? 1 ?? + 1 2 ?? ?? 2 = 0 + 1 2 ( 12) ( 5) 2 = 150rad . Q 3. If the position vector of a particle is ??? ? = ( ?? ??ˆ + ?? ??ˆ) meter and its angular velocity is ??? ? ?? = ( ??ˆ + ?? ?? ˆ ) ?????? /?????? then its linear velocity is (in ?? /?? ) (a) ( ?? ??ˆ - ?? ??ˆ + ?? ?? ˆ ) (b) ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) (c) -( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) (d) ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) Ans: (a) ?? ? ? = ?? ? ? ? × ?? ? = ( 3??ˆ + 4??ˆ + 0?? ˆ )× ( 0??ˆ + ??ˆ + 2?? ˆ ) = | ??ˆ ??ˆ ?? ˆ 3 4 0 0 1 2 | = 8??ˆ - 6??ˆ + 3?? ˆ Q 4. A circular disc ?? of radius ?? is made from an iron plate of thickness ?? , and another disc ?? of radius ?? ?? is made from an iron plate of thickness ?? ?? . Then the relation between the moment of inertia ?? ?? and ?? ?? is (a) ?? ?? = ???? ?? ?? (b) ?? ?? = ???? ?? ?? (c) ?? ?? = ???? ?? ?? (d) ?? ?? = ?? ?? Ans: (a) Moment of Inertia of disc ?? = 1 2 ?? ?? 2 = 1 2 ( ?? ?? 2 ???? ) ?? 2 = 1 2 ?????? ?? 4 [As ?? = ?? × ?? = ?? ?? 2 ???? where ?? = thickness, ?? = density] ? ?? ?? ?? ?? = ?? ?? ?? ?? ( ?? ?? ?? ?? ) 4 ? ?? ?? ?? ?? = 1 4 ( 4) 4 = 64 ? ?? ?? = 64?? ?? [If ?? = constant] [Given ?? ?? = 4?? ?? , ?? ?? = ?? ?? 4 ] Q 5. Three rings each of mass ?? and radius ?? are arranged as shown in the figure. The moment of inertia of the system about ?? ?? ' will be (a) ?? ?? ?? ?? (b) ?? ?? ?? ?? ?? (c) ?? ?? ?? ?? (d) ?? ?? ?? ?? ?? Ans: (d) M.I of system about ?? ?? ' ?? = ?? 1 + ?? 2 + ?? 3 where ?? 1 = moment of inertia of ring about diameter, ?? 2 = ?? 3 = M.I. of inertia of ring about a tangent in a plane ? ?? = 1 2 ?? ?? 2 + 3 2 ?? ?? 2 + 3 2 ?? ?? 2 = 7 2 ?? ?? 2 Q 6. The moment of inertia of a solid sphere of density ?? and radius ?? about its diameter is (a) ?????? ?????? ?? ?? ?? (b) ?????? ?????? ?? ?? ?? (c) ?????? ?????? ?? ?? ?? (d) ?????? ?????? ?? ?? ?? Ans: (c) Moment of inertia of sphere about it diameter ?? = 2 5 ?? ?? 2 = 2 5 ( 4 3 ?? ?? 3 ?? )?? 2 [As ?? = ???? = 4 3 ?? ?? 3 ?? ] ?? = 8?? 15 ?? 5 ?? = 8 × 22 15 × 7 ?? 5 ?? = 176 105 ?? 5 ?? Q 7. A force of ( ?? ??ˆ - ?? ??ˆ + ?? ?? ˆ ) ?? acts at a point ( ?? ??ˆ + ?? ??ˆ - ?? ?? ˆ ) metre from the origin. The magnitude of torque is (a) Zero (b) ???? . ?? ?? - ?? (c) ?? . ?????? ?? - ?? (d) ?? . ?????? ?? - ?? Ans: (b) ?? ? = ( 2??ˆ - 4??ˆ + 2?? ˆ ) ?? and ?? ? = ( 3?? + 2 - 4?? ˆ ) meter Torque ?? ? = ?? ? × ?? ? = | ??ˆ ??ˆ ?? ˆ 3 2 -4 2 -4 2 | ? ?? ? = -12??ˆ - 14??ˆ - 16?? ˆ and |?? ?| = v( -12) 2 + ( -14) 2 + ( -16) 2 = 24.4 ?? - ?? Page 4 Solved Examples on Rotational Motion JEE Mains Single Correct Q1. The velocities of three particles of masses ???? ?? , ???? ?? and ???? ?? are ???? ???, ???? ???, and ???? ?? ? ? ? respectively. The velocity of the centre of mass of the three particles is (a) ?? ??? + ?? ??? + ?? ?? ? ? ? (b) ???? ( ??? + ??? + ?? ? ? ? ) (c) ???? ??? + ???? ??? + ?? ?? ? ? ? (d) ?? ??? + ???? ??? + ???? ?? ? ? ? Ans : (a) Velocity of centre of mass ?? ???? = ?? 1 ?? 1 +?? 2 ?? 2 +?? 3 ?? 3 ?? 1 +?? 2 +?? 3 = 20×10??ˆ+30×10?? ˆ+50×10?? ˆ 100 = 2??ˆ + 3??ˆ + 5?? ˆ . Q1. Angular displacement ( ?? ) of a flywheel varies with time as ?? = ???? + ?? ?? ?? + ?? ?? ?? then angular acceleration is given by (a) ?? + ?? ???? - ?? ?? ?? ?? (b) ?? ?? - ?? ?? (c) ?? + ?? ?? - ?? ?? (d) ?? ?? + ?? ???? Ans: (d) Angular acceleration ?? = ?? 2 ?? ?? ?? 2 = ?? 2 ?? ?? 2 ( ???? + ?? ?? 2 + ?? ?? 3 ) = 2?? + 6???? Q 2. A wheel is at rest. Its angular velocity increases uniformly and becomes ?????????? /?????? after 5 sec. The total angular displacement is (a) ???????????? (b) ?????????? (c) ???????????? (d) ???????????? Ans: (d) Angular acceleration ?? = ?? 2 -?? 1 ?? = 60-0 5 = 12rad /sec 2 Now from ?? = ?? 1 ?? + 1 2 ?? ?? 2 = 0 + 1 2 ( 12) ( 5) 2 = 150rad . Q 3. If the position vector of a particle is ??? ? = ( ?? ??ˆ + ?? ??ˆ) meter and its angular velocity is ??? ? ?? = ( ??ˆ + ?? ?? ˆ ) ?????? /?????? then its linear velocity is (in ?? /?? ) (a) ( ?? ??ˆ - ?? ??ˆ + ?? ?? ˆ ) (b) ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) (c) -( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) (d) ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) Ans: (a) ?? ? ? = ?? ? ? ? × ?? ? = ( 3??ˆ + 4??ˆ + 0?? ˆ )× ( 0??ˆ + ??ˆ + 2?? ˆ ) = | ??ˆ ??ˆ ?? ˆ 3 4 0 0 1 2 | = 8??ˆ - 6??ˆ + 3?? ˆ Q 4. A circular disc ?? of radius ?? is made from an iron plate of thickness ?? , and another disc ?? of radius ?? ?? is made from an iron plate of thickness ?? ?? . Then the relation between the moment of inertia ?? ?? and ?? ?? is (a) ?? ?? = ???? ?? ?? (b) ?? ?? = ???? ?? ?? (c) ?? ?? = ???? ?? ?? (d) ?? ?? = ?? ?? Ans: (a) Moment of Inertia of disc ?? = 1 2 ?? ?? 2 = 1 2 ( ?? ?? 2 ???? ) ?? 2 = 1 2 ?????? ?? 4 [As ?? = ?? × ?? = ?? ?? 2 ???? where ?? = thickness, ?? = density] ? ?? ?? ?? ?? = ?? ?? ?? ?? ( ?? ?? ?? ?? ) 4 ? ?? ?? ?? ?? = 1 4 ( 4) 4 = 64 ? ?? ?? = 64?? ?? [If ?? = constant] [Given ?? ?? = 4?? ?? , ?? ?? = ?? ?? 4 ] Q 5. Three rings each of mass ?? and radius ?? are arranged as shown in the figure. The moment of inertia of the system about ?? ?? ' will be (a) ?? ?? ?? ?? (b) ?? ?? ?? ?? ?? (c) ?? ?? ?? ?? (d) ?? ?? ?? ?? ?? Ans: (d) M.I of system about ?? ?? ' ?? = ?? 1 + ?? 2 + ?? 3 where ?? 1 = moment of inertia of ring about diameter, ?? 2 = ?? 3 = M.I. of inertia of ring about a tangent in a plane ? ?? = 1 2 ?? ?? 2 + 3 2 ?? ?? 2 + 3 2 ?? ?? 2 = 7 2 ?? ?? 2 Q 6. The moment of inertia of a solid sphere of density ?? and radius ?? about its diameter is (a) ?????? ?????? ?? ?? ?? (b) ?????? ?????? ?? ?? ?? (c) ?????? ?????? ?? ?? ?? (d) ?????? ?????? ?? ?? ?? Ans: (c) Moment of inertia of sphere about it diameter ?? = 2 5 ?? ?? 2 = 2 5 ( 4 3 ?? ?? 3 ?? )?? 2 [As ?? = ???? = 4 3 ?? ?? 3 ?? ] ?? = 8?? 15 ?? 5 ?? = 8 × 22 15 × 7 ?? 5 ?? = 176 105 ?? 5 ?? Q 7. A force of ( ?? ??ˆ - ?? ??ˆ + ?? ?? ˆ ) ?? acts at a point ( ?? ??ˆ + ?? ??ˆ - ?? ?? ˆ ) metre from the origin. The magnitude of torque is (a) Zero (b) ???? . ?? ?? - ?? (c) ?? . ?????? ?? - ?? (d) ?? . ?????? ?? - ?? Ans: (b) ?? ? = ( 2??ˆ - 4??ˆ + 2?? ˆ ) ?? and ?? ? = ( 3?? + 2 - 4?? ˆ ) meter Torque ?? ? = ?? ? × ?? ? = | ??ˆ ??ˆ ?? ˆ 3 2 -4 2 -4 2 | ? ?? ? = -12??ˆ - 14??ˆ - 16?? ˆ and |?? ?| = v( -12) 2 + ( -14) 2 + ( -16) 2 = 24.4 ?? - ?? Q 8. The position of a particle is given by : ??? ? = ( ??ˆ + ?? ??ˆ - ?? ˆ ) and momentum ?? ? ?? = ( ?? ??ˆ + ?? ??ˆ - ?? ?? ˆ ) . The angular momentum is perpendicular to (a) ?? -axis (b) ?? -axis (c) ?? -axis (d) Line at equal angles to all the three axes Ans: (a) ?? ? ? = ?? ? × ?? ? = | ??ˆ ??ˆ ?? ˆ 1 2 -1 3 4 -2 | = 0??ˆ - ??ˆ - 2?? ˆ = -??ˆ - 2?? ˆ and the ?? - axis is given by ?? + 0??ˆ + 0?? ˆ Dot product of these two vectors is zero i.e. angular momentum is perpendicular to ?? -axis. Q 9. A solid cylinder of mass ?? ???? and radius ?? . ?? ?? is rotating about its own axis without friction with angular velocity ???????? /?? . A particle of mass ?? . ?? ???? and moving with a velocity ?? ?? /?? strikes the cylinder and sticks to it as shown in figure. The angular momentum of the cylinder before collision will be (a) ?? . ???? ?? - ?? (b) ???? ?? - ?? (c) ?? . ?? ?? - ?? (d) ?? . ???? ?? - ?? Ans: (a) Angular momentum of the cylinder before collision ?? = ???? = 1 2 ?? ?? 2 ?? = 1 2 ( 2) ( 0.2) 2 × 3 = 0.12 J-s. Q 10. In the above problem the angular velocity of the system after the particle sticks to it will be (a) ?? . ???????? /?? (b) ?? . ???????? /?? (c) ???? . ???????? /?? (d) ???? . ???????? /?? Ans: (c) Initial angular momentum of bullet + initial angular momentum of cylinder = Final angular momentum of (bullet + cylinder) system ? ?????? + ?? 1 ?? = ( ?? 1 + ?? 2 ) ?? ' ? ?????? + ?? 1 ?? = ( 1 2 ?? ?? 2 + ?? ?? 2 )?? ' ? 0.5 × 5 × 0.2 + 0.12 = ( 1 2 2( 0.2) 2 + ( 0.5) ( 0.2) 2 )?? ' ? ?? ' = 10.3rad /sec. Page 5 Solved Examples on Rotational Motion JEE Mains Single Correct Q1. The velocities of three particles of masses ???? ?? , ???? ?? and ???? ?? are ???? ???, ???? ???, and ???? ?? ? ? ? respectively. The velocity of the centre of mass of the three particles is (a) ?? ??? + ?? ??? + ?? ?? ? ? ? (b) ???? ( ??? + ??? + ?? ? ? ? ) (c) ???? ??? + ???? ??? + ?? ?? ? ? ? (d) ?? ??? + ???? ??? + ???? ?? ? ? ? Ans : (a) Velocity of centre of mass ?? ???? = ?? 1 ?? 1 +?? 2 ?? 2 +?? 3 ?? 3 ?? 1 +?? 2 +?? 3 = 20×10??ˆ+30×10?? ˆ+50×10?? ˆ 100 = 2??ˆ + 3??ˆ + 5?? ˆ . Q1. Angular displacement ( ?? ) of a flywheel varies with time as ?? = ???? + ?? ?? ?? + ?? ?? ?? then angular acceleration is given by (a) ?? + ?? ???? - ?? ?? ?? ?? (b) ?? ?? - ?? ?? (c) ?? + ?? ?? - ?? ?? (d) ?? ?? + ?? ???? Ans: (d) Angular acceleration ?? = ?? 2 ?? ?? ?? 2 = ?? 2 ?? ?? 2 ( ???? + ?? ?? 2 + ?? ?? 3 ) = 2?? + 6???? Q 2. A wheel is at rest. Its angular velocity increases uniformly and becomes ?????????? /?????? after 5 sec. The total angular displacement is (a) ???????????? (b) ?????????? (c) ???????????? (d) ???????????? Ans: (d) Angular acceleration ?? = ?? 2 -?? 1 ?? = 60-0 5 = 12rad /sec 2 Now from ?? = ?? 1 ?? + 1 2 ?? ?? 2 = 0 + 1 2 ( 12) ( 5) 2 = 150rad . Q 3. If the position vector of a particle is ??? ? = ( ?? ??ˆ + ?? ??ˆ) meter and its angular velocity is ??? ? ?? = ( ??ˆ + ?? ?? ˆ ) ?????? /?????? then its linear velocity is (in ?? /?? ) (a) ( ?? ??ˆ - ?? ??ˆ + ?? ?? ˆ ) (b) ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) (c) -( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) (d) ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) Ans: (a) ?? ? ? = ?? ? ? ? × ?? ? = ( 3??ˆ + 4??ˆ + 0?? ˆ )× ( 0??ˆ + ??ˆ + 2?? ˆ ) = | ??ˆ ??ˆ ?? ˆ 3 4 0 0 1 2 | = 8??ˆ - 6??ˆ + 3?? ˆ Q 4. A circular disc ?? of radius ?? is made from an iron plate of thickness ?? , and another disc ?? of radius ?? ?? is made from an iron plate of thickness ?? ?? . Then the relation between the moment of inertia ?? ?? and ?? ?? is (a) ?? ?? = ???? ?? ?? (b) ?? ?? = ???? ?? ?? (c) ?? ?? = ???? ?? ?? (d) ?? ?? = ?? ?? Ans: (a) Moment of Inertia of disc ?? = 1 2 ?? ?? 2 = 1 2 ( ?? ?? 2 ???? ) ?? 2 = 1 2 ?????? ?? 4 [As ?? = ?? × ?? = ?? ?? 2 ???? where ?? = thickness, ?? = density] ? ?? ?? ?? ?? = ?? ?? ?? ?? ( ?? ?? ?? ?? ) 4 ? ?? ?? ?? ?? = 1 4 ( 4) 4 = 64 ? ?? ?? = 64?? ?? [If ?? = constant] [Given ?? ?? = 4?? ?? , ?? ?? = ?? ?? 4 ] Q 5. Three rings each of mass ?? and radius ?? are arranged as shown in the figure. The moment of inertia of the system about ?? ?? ' will be (a) ?? ?? ?? ?? (b) ?? ?? ?? ?? ?? (c) ?? ?? ?? ?? (d) ?? ?? ?? ?? ?? Ans: (d) M.I of system about ?? ?? ' ?? = ?? 1 + ?? 2 + ?? 3 where ?? 1 = moment of inertia of ring about diameter, ?? 2 = ?? 3 = M.I. of inertia of ring about a tangent in a plane ? ?? = 1 2 ?? ?? 2 + 3 2 ?? ?? 2 + 3 2 ?? ?? 2 = 7 2 ?? ?? 2 Q 6. The moment of inertia of a solid sphere of density ?? and radius ?? about its diameter is (a) ?????? ?????? ?? ?? ?? (b) ?????? ?????? ?? ?? ?? (c) ?????? ?????? ?? ?? ?? (d) ?????? ?????? ?? ?? ?? Ans: (c) Moment of inertia of sphere about it diameter ?? = 2 5 ?? ?? 2 = 2 5 ( 4 3 ?? ?? 3 ?? )?? 2 [As ?? = ???? = 4 3 ?? ?? 3 ?? ] ?? = 8?? 15 ?? 5 ?? = 8 × 22 15 × 7 ?? 5 ?? = 176 105 ?? 5 ?? Q 7. A force of ( ?? ??ˆ - ?? ??ˆ + ?? ?? ˆ ) ?? acts at a point ( ?? ??ˆ + ?? ??ˆ - ?? ?? ˆ ) metre from the origin. The magnitude of torque is (a) Zero (b) ???? . ?? ?? - ?? (c) ?? . ?????? ?? - ?? (d) ?? . ?????? ?? - ?? Ans: (b) ?? ? = ( 2??ˆ - 4??ˆ + 2?? ˆ ) ?? and ?? ? = ( 3?? + 2 - 4?? ˆ ) meter Torque ?? ? = ?? ? × ?? ? = | ??ˆ ??ˆ ?? ˆ 3 2 -4 2 -4 2 | ? ?? ? = -12??ˆ - 14??ˆ - 16?? ˆ and |?? ?| = v( -12) 2 + ( -14) 2 + ( -16) 2 = 24.4 ?? - ?? Q 8. The position of a particle is given by : ??? ? = ( ??ˆ + ?? ??ˆ - ?? ˆ ) and momentum ?? ? ?? = ( ?? ??ˆ + ?? ??ˆ - ?? ?? ˆ ) . The angular momentum is perpendicular to (a) ?? -axis (b) ?? -axis (c) ?? -axis (d) Line at equal angles to all the three axes Ans: (a) ?? ? ? = ?? ? × ?? ? = | ??ˆ ??ˆ ?? ˆ 1 2 -1 3 4 -2 | = 0??ˆ - ??ˆ - 2?? ˆ = -??ˆ - 2?? ˆ and the ?? - axis is given by ?? + 0??ˆ + 0?? ˆ Dot product of these two vectors is zero i.e. angular momentum is perpendicular to ?? -axis. Q 9. A solid cylinder of mass ?? ???? and radius ?? . ?? ?? is rotating about its own axis without friction with angular velocity ???????? /?? . A particle of mass ?? . ?? ???? and moving with a velocity ?? ?? /?? strikes the cylinder and sticks to it as shown in figure. The angular momentum of the cylinder before collision will be (a) ?? . ???? ?? - ?? (b) ???? ?? - ?? (c) ?? . ?? ?? - ?? (d) ?? . ???? ?? - ?? Ans: (a) Angular momentum of the cylinder before collision ?? = ???? = 1 2 ?? ?? 2 ?? = 1 2 ( 2) ( 0.2) 2 × 3 = 0.12 J-s. Q 10. In the above problem the angular velocity of the system after the particle sticks to it will be (a) ?? . ???????? /?? (b) ?? . ???????? /?? (c) ???? . ???????? /?? (d) ???? . ???????? /?? Ans: (c) Initial angular momentum of bullet + initial angular momentum of cylinder = Final angular momentum of (bullet + cylinder) system ? ?????? + ?? 1 ?? = ( ?? 1 + ?? 2 ) ?? ' ? ?????? + ?? 1 ?? = ( 1 2 ?? ?? 2 + ?? ?? 2 )?? ' ? 0.5 × 5 × 0.2 + 0.12 = ( 1 2 2( 0.2) 2 + ( 0.5) ( 0.2) 2 )?? ' ? ?? ' = 10.3rad /sec. Q 11. A body of moment of inertia of ?? ???? - ?? ?? rotating with an angular velocity of ???????? /?????? has the same kinetic energy as a mass of ???? ???? moving with a velocity of (a) ?? ?? /?? (b) ?? . ?? ?? /?? (c) ?? ?? /?? (d) ?? ?? /?? Ans: (d) Rotational kinetic energy of the body = 1 2 I?? 2 and translatory kinetic energy = 1 2 ?? ?? 2 According to problem = 1 2 ?? ?? 2 = 1 2 ?? ?? 2 ? 1 2 × 3 × ( 2) 2 = 1 2 × 12 × ?? 2 ? ?? = 1 m/s. Q 12. A disc and a ring of same mass are rolling and if their kinetic energies are equal, then the ratio of their velocities will be (a) v ?? : v ?? (b) v ?? : v ?? (c) v ?? : v ?? (d) v ?? : v ?? Ans: (a) ?? disc = 1 2 ?? ?? ?? 2 (1 + ?? 2 ?? 2 ) = 3 4 ?? ?? ?? 2 [ ???? ?? 2 ?? 2 = 1 2 for disc ] ?? ring = 1 2 ?? ?? ?? (1 + ?? 2 ?? 2 ) = ?? ?? ?? 2 [ ???? ?? 2 ?? 2 = 1 for ring ] According to problem ?? disc = ?? ring ? 3 4 ?? ?? ?? 2 = ?? ?? ?? 2 ? ?? ?? ?? ?? = v 4 3 . Q 13. A wheel is rotating with an angular speed of ?????????? /?????? . It is stopped to rest by applying a constant torque in ?? ?? . If the moment of inertia of the wheel about its axis is ?? . ???? ???? - ?? ?? , then the work done by the torque in two seconds will be (a) ???? ?? (b) ???? ?? (c) ???? ?? (d) ???? ?? Ans: (c) ?? 1 = 20rad /sec, ?? 2 = 0, ?? = 4sec. So angular retardation ?? = ?? 1 -?? 2 ?? = 20 4 = 5rad /sec 2 Now angular speed after 2sec ?? 2 = ?? 1 - ???? = 20 - 5 × 2 = 10rad /sec Work done by torque in 2sec = loss in kinetic energy = 1 2 ?? ( ?? 1 2 - ?? 2 2 )= 1 2 ( 0.20) ( ( 20) 2 - ( 10) 2 ) = 1 2 × 0.2 × 300 = 30 J. Q 14. A ring solid sphere and a disc are rolling down from the top of the same height, then the sequence to reach on surface is (a) Ring, disc, sphereRead More
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Rotation Motion Solved Examples Notes
Rotation Motion Solved Examples Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Rotation Motion Solved Examples.
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Rotation Motion Solved Examples JEE Questions
The "Rotation Motion Solved Examples JEE Questions" guide is a valuable resource for all aspiring students preparing for the
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The content of Rotation Motion Solved Examples is prepared as per the latest JEE syllabus.
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