Lecture 3: Forces and Dynamics

Dynamics

Dynamics

  • Deals with the study of forces and their effect on motion.

Newton's Second Law

  • F = ma is the fundamental equation.
    • F represents force.
    • m represents mass.
    • a represents acceleration.
  • Applies to a point mass or an object that can be treated as a point mass.

Non-inertial Reference Frames

  • Lab reference frame is considered a non-inertial reference frame.
  • In a non-inertial reference frame, the equation F_{net} = ma does not hold true directly. A fictitious force needs to be introduced.
  • Fictitious Force: F_{fictitious} = -mA, where A is the acceleration of the non-inertial reference frame.
  • The transformation of acceleration from an inertial frame to a non-inertial frame is given by: a = a' + A, where:
    • a is the acceleration in the inertial frame.
    • a' is the acceleration in the non-inertial frame.
    • A is the acceleration of the non-inertial frame with respect to the inertial frame.
  • Therefore, a' = a + (-A) = a - A
  • The net force in the inertial frame is F_{net} = m(a'+ A)
  • In the non-inertial frame, the modified Newton's second law is:
    • F_{net} + (-mA) = ma'
    • F{net} + F{fictitious} = ma'
    • If one were to use the non-intertial reference frame, then F_{net} = ma'

Example: Pendulum in an Accelerating Car

  • Consider a pendulum inside a car accelerating with acceleration A.
  • The tension in the string is T.
  • In the non-inertial frame (inside the car), the forces acting on the pendulum bob are:
    • Tension T.
    • Weight mg.
    • Fictitious force -mA.
  • The pendulum will deflect at an angle such that the net force balances.
  • T_x = mA
  • T_y = mg
  • The time period of the pendulum is T = 2\pi \sqrt{\frac{l}{g{eff}}}, where g{eff} is the effective acceleration due to gravity.
  • Here, g_{eff} = \sqrt{g^2 + A^2}

Steps to Solve F = ma Problems

  • Applicable to n-object systems.
  • Write down F = ma for each object. This will give you n equations.
  • Identify all unknown variables (n+m in total).
  • Identify all forces acting on each object.
  • Identify constraints on the motion of the objects. These constraints will give you m additional equations.
  • Solve the system of equations to find the unknowns.

Example: Two Blocks Connected by a String

  • Two blocks with masses M and m are connected by a string over a pulley.
  • M is on a horizontal surface, and m is hanging vertically.
  • Forces on M: Tension T and friction.
  • Forces on m: Tension T and weight mg.
  • Equations for M:
    • \sum Fx = T = Max
    • \sum F_y = N - Mg = 0
  • Equations for m:
    • \sum Fy = mg - T = may
  • Constraint: ax = ay = a
  • Solve for T and a.

Example: Block on an Inclined Plane in an Accelerating System

  • A block of mass m is on an inclined plane (angle θ) inside an accelerating system (acceleration Ax). The acceleration in y direction is zero (Ay = 0).
  • The acceleration in the non-inertial frame is a = a' + A
  • ax = a'x + (-A_x)
  • ay = a'y + 0
  • Forces: Normal force N, weight mg.
  • Equations:
    • N \sin(\theta) = m a_x (∑Fx = max)
    • N \cos(\theta) = mg (∑Fy = may)
  • Constraint Example: Two masses connected by a string over a pulley.
  • m1 and m2 are connected by the string the constraints that relate the acceleration is a1 = -a2
  • m1 a1 = T - m_1g
  • m2 a2 = m_2g -T
  • \frac{d^2x1}{dt^2} = - \frac{d^2x2}{dt^2}