System of Particles

General Instructions

Full Marks: 50

1. Total numbers of questions are 20.
2. All questions are compulsory.
3. This question paper consists of four sections. :
   Section – A, Section – B, Section – C & Section – D.
4. Section – A contains 5 questions each carrying 1 marks.
5. Section – B contains 6 questions each carrying 2 marks.
6. Section – C contains 6 questions each carrying 3 marks.
7. Section – D contains 3 questions each carrying 5 marks.
8. Write answers in sections.
9. No students shall be allowed for leave the examination hall before the completion of the exam.

Section A

MCQ test

Section B

1. Define center of mass.
2. If \(m_1\), \(m_2\) and \(m_3\) are masses located at positions \(\vec r_1\), \(\vec r_2\) and \(\vec r_3\) respectively. Write the position vector of its center of mass.
3. What do you mean by a couple? Give a suitable example.
4. Make a suitable diagram of a lever showing load, load arm, effort, effort arm and fulcrum.
5. Write all the kinematics equations involving circular motion.
6. In Diwali, a boy throws a lighted bomb towards the sky. The trajectory of the bomb was \(y^2 = 4ax – 15\). After 10 seconds when the bomb was at a height of 3 m, it bursts. Find the new trajectory of the bomb.

Section C

1. If two weights of mass 5 kg and 15 kg are placed along x-axis at a distance of 3 m and 1 m respectively from the origin, find its center of the mass.
2. Find the value of \(\vec a \times \vec b \) if \(\vec a\) = \(5 \hat i + 3 \hat j + \hat k\) and \(\vec b\) = \(\hat i+ 7 \hat j + 8 \hat k\).
3. What do you mean by torque? Name the factors, torque depends upon.
4. A force \(\vec F = 7 \hat i – 10 \hat j \) N acts at a particle positioned at \(\vec r = 2 \hat i+ 3 \hat j \) about the z-axis. Find the torque.
5. Three particles lie at the vertices of an equilateral triangle. Their masses are 2 kg, 3 kg, and 4 kg respectively. Find the center of mass if each side of the equilateral triangle is 0.8 m long. (refer – NCERT ex 6.1)
6. Define moment of inertia. Find the moment of inertia of a hollow cylinder of radius 0.35 m and mass 5 kg about its axis.

Section D

1. Derive the equation for work done by a torque. Also derive the equation for power delivered by it.
2. Question on mechanical equilibrium.
   1. What do you mean by mechanical equilibrium. Write its mathematical expression.
   2. A ladder of mass 15 kg and length 4 m leans on a friction-less wall. Its feet rest on the floor 1.5 m from the wall. What will be the reaction forces of the wall and the floor on the ladder. (refer – NCERT ex 6.9).
3. Miscellaneous questions
   1. What do you mean by radius of gyration?
   2. A boy is standing at the center of a turntable. His hands are folded. What happens to the angular speed of the boy if it unfolds his arms?
   3. What are the centers of mass of a cylinder and a sphere, each of uniform density. Is it necessary that center of mass will always lie inside the body? Give example in support of your answer.