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:
  • $F_{net} + (-mA) = ma'$
  • $F<em>{net} + F</em>{fictitious} = ma'$
  • 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 $T = 2\pi \sqrt{\frac{l}{g<em>{eff}}}$, where $g</em>{eff}$ 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:
  • $\sum F<em>x = T = Ma</em>x$
  • $\sum F_y = N - Mg = 0$
• Equations for m:
  • $\sum F<em>y = mg - T = ma</em>y$
• Constraint: $a<em>x = a</em>y = a$
• 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 $A<em>x$). The acceleration in y direction is zero ($A</em>y = 0$).
• The acceleration in the non-inertial frame is $a = a' + A$
• $a<em>x = a'</em>x + (-A_x)$
• $a<em>y = a'</em>y + 0$
• 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 $a1 = -a2$
• $m<em>1 a</em>1 = T - m_1g$
• $m<em>2 a</em>2 = m_2g -T$
• $\frac{d^2x<em>1}{dt^2} = - \frac{d^2x</em>2}{dt^2}$