Newton's Laws of Motion and Applications

One-Dimensional and Two-Dimensional Kinematics

Understanding the motion of objects in one or more dimensions is fundamental in physics. The position of an object as a function of time can be described using equations for its coordinates, and the net force acting on the object can be determined using Newton's Second Law.

• Position Functions: The position of a mass in the x and y directions can be given as functions of time, such as and (where x and y are in meters, t in seconds).

• Velocity: The velocity components are found by differentiating the position functions with respect to time:

• Acceleration: The acceleration components are the second derivatives:

• Net Force: The net force acting on a mass is given by Newton's Second Law:

• Magnitude of Net Force: For two-dimensional motion:

Example: For a 3.00-kg mass with and , find the net force at s by calculating and , then using the formula above.

Statics: Equilibrium and Forces on a Wire

When objects are in equilibrium, the sum of all forces and torques acting on them is zero. This principle is used to analyze forces in systems such as wires, beams, and pulleys.

• Equilibrium Condition: and

• Application: When a bird lands on a wire, the wire sags, creating tension forces that balance the bird's weight.

• Free-Body Diagram: Draw all forces acting on the bird and the wire at the point of contact.

• Vertical Force Balance: The vertical components of the tension in the wire must sum to the bird's weight.

• Horizontal Force Balance: The horizontal components of the tension must be equal and opposite.

Example: For a 1.00-kg bird causing a 0.090 m sag in a 5.00 m span, calculate the tension in the wire using trigonometry and equilibrium conditions.

Spring Scales and Pulley Systems in Equilibrium

Spring scales measure force (in newtons) and can be used in systems with pulleys and masses. When the system is in equilibrium, the readings reflect the forces transmitted through the strings.

• Spring Scale Reading: In equilibrium, the scale reads the tension in the string.

• Pulley Systems: If pulleys and strings are massless and frictionless, the tension is the same throughout the string.

• Force Balance: The sum of forces on each mass must be zero in equilibrium.

Example: For a system with two masses and spring scales, the reading equals the weight of the hanging mass ().

Newton's Second Law: Applications and Accelerometers

Accelerometer Construction and Analysis

An accelerometer can be constructed using a mass suspended from a string. When the system accelerates, the string makes an angle with the vertical, which can be analyzed using Newton's laws.

• Forces on the Mass: The tension in the string has vertical and horizontal components.

• Angle with Vertical: If the cart accelerates horizontally with acceleration , the angle satisfies:

• Expression for Acceleration:

Example: If , m/s2.

Two-Mass Systems and Tension

When two masses are connected by a string and a force is applied, both masses accelerate together. The acceleration and tension can be found using Newton's Second Law.

• System Acceleration:

• Tension in the String: (if is applied to and is being pulled)

Example: For and on a frictionless surface with force applied, calculate and using the formulas above.

Forces on a Boat: Multiple Pulls

When multiple forces act on an object, the net force determines the acceleration according to Newton's Second Law.

• Same Direction: Add the forces to find the net force.

• Opposite Directions: Subtract the smaller force from the larger to find the net force.

• Acceleration:

Example: For a 209 kg boat, if the net force is , then m/s2.

Summary Table: Key Equations and Concepts