Centre of Mass Quiz

Centre of Mass

• The centre of mass of a system of particles is a point where the total mass of the system can be considered to be concentrated.
• For two particles of equal mass (m1 = m2 = m), the centre of mass lies exactly midway between them.
• For n particles with different masses (m1, m2, … mn) along a straight line, the position of the centre of mass is given by the following formula:
  • X = (m1x1 + m2x2 + … + mnxn) / (m1 + m2 + … + mn)
• For a system of n particles in space, the coordinates of the centre of mass are given by:
  • X = (∑ mi xi) / M
  • Y = (∑ mi yi) /M
  • Z = (∑ mi zi) / M
• The total mass of the system is given by M = ∑ mi.
• The position of the ith particle is given by (xi, yi, zi).
• These equations are applicable to a rigid body, which can be treated as a continuous distribution of mass when the number of particles is very large.

Motion of the Centre of Mass

• Newton’s second law of motion states that the force acting on a particle is equal to its mass times its acceleration. (F = ma)
• The force acting on a system of particles can be calculated by summing up the forces acting on each individual particle.
• The acceleration of the centre of mass (A) of a system of particles is given by:
  • A = (∑ mi ai) / M
• The total force on the system (F) is the sum of the forces acting on individual particles:
  • F = F1 + F2 + … + Fn
  • Where F1, F2, … Fn are the forces acting on the first, second, … nth particle, respectively.
• Therefore, applying Newton’s second law to the entire system:
  • MA = F1 + F2 + … + Fn

Linear Momentum

• The linear momentum of a particle is its mass times its velocity (p = mv).
• The total linear momentum of a system of particles is the vector sum of the linear momenta of the individual particles.
• The total linear momentum of a system of particles is given by:
  • P = m1v1 + m2v2 + … + mnv n
  • Where v1, v2, … vn are the velocities of the first, second, … nth particle, respectively.
• Applying Newton’s second law to the total linear momentum, we get:
  • dP/dt = F
• This equation expresses the conservation of linear momentum: the total linear momentum of a system remains constant if the net external force acting on the system is zero.

Conservation of Linear Momentum

• The total linear momentum of an isolated system remains constant.
• In a closed system, where no external forces act, the total linear momentum before and after an interaction remains constant.
• Examples of conservation of linear momentum include:
  • The motion of a rocket after it expels fuel.
  • The collision of two particles.
  • Radioactive decay of a nucleus.

Centre of Mass Frame of Reference

• It is often convenient to work in the centre of mass frame of reference when analysing the motion of a system of particles.
• In this frame, the centre of mass of the system is at rest.
• The motion of the particles in the centre of mass frame can be used to understand the motion of the system as a whole.
• Examples include:
  • The motion of binary stars.
  • The decay of a radioactive nucleus.

Vector Product of Two Vectors

• The vector product of two vectors is a vector quantity that is perpendicular to both of the original vectors.
• It is denoted by the symbol x and is calculated using the following formula:
  • a x b = |a||b|sinθ n
  • Where a and b are two vectors, θ is the angle between them, and n is a unit vector perpendicular to both a and b.
• The direction of the vector product is determined using the right-hand rule.

Right Hand Rule

• To apply the right-hand rule:
  • Point the fingers of your right hand in the direction of the first vector.
  • Curl your fingers towards the direction of the second vector.
  • Your thumb will now be pointing in the direction of the vector product.