1 Mathematics • Dynamics A solid homogeneous cone of height h and semi-vertical angle alpha oscillates about a diameter of its base. The length of the simple equivalent pendulum is: A solid homogeneous cone of height h and semi-vertical angle alpha oscillates about a diameter of its base. The length of the simple equivalent pendulum is: A. (h/5)(2 + 5tan^2(alpha)) (h/5)(2 + 5tan^2(alpha)) B. (h/5)(2 + 3tan^2(alpha)) (h/5)(2 + 3tan^2(alpha)) C. (h/3)(2 + 3tan^2(alpha)) (h/3)(2 + 3tan^2(alpha)) D. (h/2)(2 + 5tan^2(alpha)) (h/2)(2 + 5tan^2(alpha)) उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: B Correct Answer: B For a physical pendulum, the equivalent length is I/(Md), where I is the moment of inertia about the axis and d is the distance of the centre of mass from it. For the stated cone this reduces to (h/5)(2 + 3tan^2(alpha)). For a physical pendulum, the equivalent length is I/(Md), where I is the moment of inertia about the axis and d is the distance of the centre of mass from it. For the stated cone this reduces to (h/5)(2 + 3tan^2(alpha)).
2 Mathematics • Dynamics The moment of inertia of a uniform thin rod of mass M and length L about a perpendicular axis of rotation at its end is: The moment of inertia of a uniform thin rod of mass M and length L about a perpendicular axis of rotation at its end is: A. (1/8)ML^2 (1/8)ML^2 B. (1/2)ML^2 (1/2)ML^2 C. (1/3)ML^2 (1/3)ML^2 D. (1/4)ML^2 (1/4)ML^2 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C The moment of inertia of a uniform rod about its centre is ML^2/12. By the parallel-axis theorem, about an end it is ML^2/12 + M(L/2)^2 = ML^2/3. The moment of inertia of a uniform rod about its centre is ML^2/12. By the parallel-axis theorem, about an end it is ML^2/12 + M(L/2)^2 = ML^2/3.
3 Mathematics • Dynamics Which one of the following is a correct statement? Which one of the following is a correct statement? A. Holonomic systems are systems for which all constraints are integrable into positional constraints. Holonomic systems are systems for which all constraints are integrable into positional constraints. B. Non-holonomic systems are systems where velocities and other derivatives of position are not constraints. Non-holonomic systems are systems where velocities and other derivatives of position are not constraints. C. Holonomic systems are systems for which all constraints are non-integrable into positional constraints. Holonomic systems are systems for which all constraints are non-integrable into positional constraints. D. Non-holonomic systems are systems that have constraints that are integrable into positional constraints. Non-holonomic systems are systems that have constraints that are integrable into positional constraints. उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A Holonomic constraints can be expressed, or integrated, as relations involving generalized coordinates and time. Therefore option A is correct. Holonomic constraints can be expressed, or integrated, as relations involving generalized coordinates and time. Therefore option A is correct.
4 Mathematics • Dynamics A rough uniform board of mass m and length 2a rests on a smooth horizontal plane. A man of mass M walks on it from one end to the other. The distance through which the board moves in this time is: A rough uniform board of mass m and length 2a rests on a smooth horizontal plane. A man of mass M walks on it from one end to the other. The distance through which the board moves in this time is: A. 3Ma/(m+M) 3Ma/(m+M) B. 3Ma/(m+2M) 3Ma/(m+2M) C. 2Ma/(m+M) 2Ma/(m+M) D. Ma/(m+M) Ma/(m+M) उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C With no external horizontal force, the centre of mass remains fixed. If the man moves 2a relative to the board, the board displacement has magnitude 2Ma/(m+M). With no external horizontal force, the centre of mass remains fixed. If the man moves 2a relative to the board, the board displacement has magnitude 2Ma/(m+M).
5 Mathematics • Dynamics The moment of inertia of an ellipse of mass M and semiaxes a and b about a tangent is, where p is the perpendicular from the centre to the tangent: The moment of inertia of an ellipse of mass M and semiaxes a and b about a tangent is, where p is the perpendicular from the centre to the tangent: A. (5M/4)p^2 (5M/4)p^2 B. (5M/6)p^2 (5M/6)p^2 C. (5M/2)p^2 (5M/2)p^2 D. (5M/3)p^2 (5M/3)p^2 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A Using the moment of inertia of the elliptic lamina about a parallel central axis together with the parallel-axis theorem gives (5M/4)p^2 for the stated tangent. Using the moment of inertia of the elliptic lamina about a parallel central axis together with the parallel-axis theorem gives (5M/4)p^2 for the stated tangent.
6 Mathematics • Dynamics A particle starts from the origin. The components of its velocity parallel to the coordinate axes at time t are 2t+3 and 4t. The path travelled by the particle is: A particle starts from the origin. The components of its velocity parallel to the coordinate axes at time t are 2t+3 and 4t. The path travelled by the particle is: A. 4x^2-y^2-4xy+18y=0 4x^2-y^2-4xy+18y=0 B. x^2+y^2+xy-12y=0 x^2+y^2+xy-12y=0 C. 3x^2+y^2+4xy-18y=0 3x^2+y^2+4xy-18y=0 D. 4x^2+y^2-4xy-18y=0 4x^2+y^2-4xy-18y=0 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: D Correct Answer: D Integration gives x=t^2+3t and y=2t^2. Eliminating t from these relations yields 4x^2+y^2-4xy-18y=0. Integration gives x=t^2+3t and y=2t^2. Eliminating t from these relations yields 4x^2+y^2-4xy-18y=0.