Particle moving in a straight line under a variable forces

Particle moving in a straight line under a variable forces

Understanding Motion in a Straight Line Under Variable Forces

  • A particle can be defined as an object of negligible size. In this context, we are considering a particle moving in a straight line under the effect of variable forces.
  • Variable forces are forces whose magnitude or direction changes with respect to time or the body’s displacement.
  • Newton’s second law states that the acceleration of an object as produced by a net force is directly proportional to the net force, in the same direction as the net force, and inversely proportional to the mass of the object (F=ma).

Applying Newton’s Second Law

  • To solve problems of particles moving under variable forces, we apply Newton’s second law.
  • This law provides a differential equation relating force (F), mass (m) and acceleration (a).
  • Remember acceleration is the derivative of velocity with respect to time, and velocity is the derivative of displacement with respect to time.

Solving by Integration

  • These problems often require us to integrate the equation derived from Newton’s second law.
  • If a force (F) is given as a function of time (t) or displacement (s), we must integrate this function to find the velocity (v) or displacement respectively.
  • Initial conditions, such as initial velocities or displacements, are used to find the constant of integration.

Energy Methods and Work Done

  • Another way to solve these problems is by using energy methods, mainly involving the concept of work done.
  • Work done by a force in moving an object is the force multiplied by the displacement in the direction of the force.
  • The Principle of Work and Energy states that the work done on a particle is equal to the change in its kinetic energy.

Key Points to Remember

  • Being able to set up and solve simple differential equations is crucial for managing variable force problems.
  • Knowing how to calculate work done by an integration of force over a displacement will be helpful during revision.
  • Understanding how to apply energy methods could make complex problems easier to solve.
  • Mastery of these techniques will be solid preparation for successfully tackling real exam problems involving particles moving under variable forces.