CH5-Newton's Laws of Motion: Equilibrium, Forces, and FBD - EXAM 2

Newton's Laws of Motion

Introduction to Newton's Laws

Newton's three laws of motion, first published in Philosophiæ Naturalis Principia Mathematica (1687), form the foundation of classical mechanics. These laws describe the relationship between the motion of an object and the forces acting upon it.

• First Law (Law of Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.

• Second Law: The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass. $\vec{F}_{\text{net}} = m\vec{a}$

• Third Law: For every action, there is an equal and opposite reaction.

These laws are essential for analyzing mechanical systems and predicting the motion of objects under various force conditions.

Equilibrium and Free-Body Diagrams (FBDs)

2D Equilibrium Example

When analyzing systems in equilibrium (constant velocity, $\vec{a} = 0$), the net force on each object is zero. Free-body diagrams (FBDs) are used to represent all forces acting on each object.

• Key Point: For a system of connected objects (e.g., a cart and bucket), draw separate FBDs for each object.

• Coordinate System: You may choose different coordinate systems for each FBD to simplify calculations.

• Example: If a cart and bucket move at constant velocity up a ramp, the net force on each is zero. The tension in the cable equals the weight of the bucket, and the weights are related by the geometry of the ramp.

Equations:

• For the bucket: $T - w_2 = 0 \implies T = w_2$

• For the cart (along the ramp): $-w_1 \sin 15^\circ = w_2 - w_1 \sin 15^\circ = 0 \implies w_2 = w_1 \sin 15^\circ$

• Normal force on the cart: $n - w_1 \cos 15^\circ = 0 \implies n = 0.97 w_1$

Constant Velocity and Net Force

When an object moves at constant velocity, its acceleration is zero, and the net force acting on it is zero. This is a direct application of Newton's First Law.

• Key Point: The sum of all force components in each direction must be zero.

• Example: For a crate on a frictionless surface with forces $\vec{F}_1$, $\vec{F}_2$, and $\vec{F}_3$, if $\vec{v}$ is constant, then $\sum F_x = 0$ and $\sum F_y = 0$.

Equations:

• $F_{1x} + F_{2x} + F_{3x} = 0$

• $F_{1y} + F_{2y} + F_{3y} = 0$

Example Calculation: If $F_{1x} = 2$ N, $F_{2x} = -5$ N, $F_{3x} = ?$, then $2 - 5 + F_{3x} = 0 \implies F_{3x} = 3$ N.

Velocity-Time Graphs and Force Diagrams

The velocity-time graph of an object can be interpreted using its FBD. If the net force is zero, velocity is constant; if the net force is nonzero, velocity changes linearly with time.

• Key Point: The direction and magnitude of forces determine the shape of the velocity-time graph.

• Example: If two equal and opposite forces act on an object, its velocity remains constant.

Applications of Newton's Laws

Elevator Problem: Tension and Acceleration

When an elevator slows down while moving downward, the tension in the supporting cable must be calculated using Newton's Second Law and kinematic equations.

• Given: Mass $m = 800$ kg, initial velocity $v_0 = -10$ m/s, displacement $\Delta y = -25$ m.

• Find: Tension $T$ in the cable as the elevator comes to rest.

Steps:

1. Use kinematics to find acceleration: $v^2 = v_0^2 + 2a_y \Delta y$

2. Apply Newton's Second Law: $\sum F_y = T - mg = ma_y$

3. Solve for $T$: $T = m(g + a_y)$

Example Calculation:

• $a_y = \frac{0^2 - (-10)^2}{2 \times (-25)} = 2.0$ m/s$^2$

• $T = 800 \times (9.8 + 2.0) = 9440$ N

Key Concepts and Definitions

• Equilibrium: A state in which the net force on an object is zero, resulting in constant velocity (which may be zero).

• Free-Body Diagram (FBD): A graphical representation showing all forces acting on an object.

• Normal Force: The perpendicular contact force exerted by a surface on an object.

• Tension: The pulling force transmitted by a string, cable, or rope.

Additional info:

• When analyzing multi-object systems, always draw separate FBDs and apply Newton's laws to each object individually.

• Choosing an appropriate coordinate system can simplify force component calculations, especially on inclined planes or ramps.