Courses

Course Description

Motion of system of particles:

A system of particles is a group of particles that move together as a whole. The motion of the system of particles is described by the motion of its center of mass. The center of mass of a system is the point at which the system can be balanced. Mathematically, the center of mass can be expressed as:

Xcm = (m1x1 + m2x2 + … + mnxn) / (m1 + m2 + … + mn)

where Xcm is the position of the center of mass, mi is the mass of the ith particle, and xi is the position of the ith particle.

The motion of the system of particles can be described using the laws of motion, which state that the acceleration of an object is proportional to the force applied to it and inversely proportional to its mass. Mathematically, the laws of motion can be expressed as:

F = ma

where F is the force applied to the system of particles, m is the total mass of the system, and a is the acceleration of the center of mass.

Rigid body:

A rigid body is a body in which the distance between any two particles of the body remains constant, regardless of the force applied to it. The motion of a rigid body can be described by the rotation and translation of the body.

Rotation of a rigid body:

The rotation of a rigid body is described by its angular displacement, velocity, and acceleration. The angular displacement is the angle through which the body rotates, and it is measured in radians. The angular velocity is the rate at which the body rotates, and it is given by:

w = d(theta) / dt

where w is the angular velocity, theta is the angular displacement, and t is the time. The angular acceleration is the rate at which the angular velocity changes, and it is given by:

alpha = dw / dt

where alpha is the angular acceleration, and w is the angular velocity.

Translation of a rigid body:

The translation of a rigid body is described by its linear displacement, velocity, and acceleration. The linear displacement is the distance traveled by a point on the body, and it is given by:

d = r(theta)

where d is the linear displacement, r is the distance between the point and the center of rotation, and theta is the angular displacement. The linear velocity is the rate at which the body moves in a straight line, and it is given by:

v = dr/dt

where v is the linear velocity, and r is the linear displacement. The linear acceleration is the rate at which the linear velocity changes, and it is given by:

a = dv/dt

where a is the linear acceleration, and v is the linear velocity.

Numerical Example:

Suppose a system of two particles of masses 2 kg and 3 kg, respectively, are placed at positions (1, 0) and (0, 1), respectively. Find the position of the center of mass of the system and the acceleration of the system when a force of 50 N is applied to it.

The position of the center of mass of the system can be found using the formula:

Xcm = (m1x1 + m2x2) / (m1 + m2)

Xcm = (2 * 1 + 3 * 0) / (2 + 3) = 0.4

The acceleration of the system can be found using the formula:

F = ma

where F is the force applied to the system, and a is the acceleration of the center of mass.

m = m1 + m2 = 2 + 3 = 5 kg

a = F / m = 50 / 5 = 10 m/s^2