Sound Intensity and Decibels

Definition and Calculation

Sound intensity is the power per unit area carried by a sound wave. The decibel (dB) scale is a logarithmic measure of sound intensity relative to a reference level.

• Sound Intensity (I): The amount of sound energy passing through a unit area per second. Units: W/m2.

• Decibel Level (L): Calculated using: where W/m2 is the threshold of hearing.

• Example: A concert speaker emits W of power uniformly in all directions. To find the intensity at 7.00 m: Then, use the decibel formula above.

Doppler Effect

Frequency Shift Due to Motion

The Doppler Effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source.

• Observed Frequency (f'): For a moving source and stationary observer: where is the speed of sound, is the speed of the source (use minus if moving toward observer).

• Example: A train whistle of frequency 1,500 Hz moves at 30 m/s. Find the observed frequency if the speed of sound is 340 m/s.

Oscillations and Springs

Mass-Spring System

A mass attached to a spring oscillates with simple harmonic motion (SHM).

• Spring Constant (k): Measures the stiffness of the spring.

• Maximum Kinetic Energy:

• Maximum Speed: where and is amplitude.

• Example: A 0.5 kg block oscillates with amplitude 0.25 m and spring constant 30 N/m. Find and .

Simple Harmonic Motion (SHM)

Equation and Properties

SHM is a type of periodic motion where the restoring force is proportional to displacement.

• General Equation:

• Angular Frequency:

• Period:

• Example: m. Find frequency, angular frequency, and period.

Simple Pendulum

Motion and Acceleration

A simple pendulum consists of a mass suspended from a string, swinging under gravity.

• Acceleration due to Gravity: (approximate for the Moon)

• Period:

• Example: A pendulum with period 1.5 s on Earth and 0.5 s on the Moon. Find acceleration due to gravity on the Moon.

Equilibrium of Forces

Force Balancing

Equilibrium occurs when the net force on an object is zero.

• Force Components: Resolve forces into x and y directions.

• Example: Two people push a box with forces of 70 N and 40 N at 60° apart. Find the net force.

Angular Momentum

Definition and Calculation

Angular momentum is the rotational analog of linear momentum.

• Formula: for a particle moving in a circle of radius .

• Angular Momentum for a Particle:

• Example: A particle of mass kg moves in a circle of diameter 2 mm at speed m/s. Find .

Rotational Dynamics

Torque and Angular Acceleration

Torque causes rotational acceleration in objects.

• Torque: where is moment of inertia, is angular acceleration.

• Moment of Inertia for Thin Ring:

• Example: A torque of 1.50 N·m is applied to a wheel (mass 1.2 kg, radius 0.35 m). Find .

Rotational Kinetic Energy

Energy in Rotating Objects

Rotational kinetic energy is the energy due to rotation.

• Formula:

• Example: A flywheel slows from 79.3 rad/s to rest, losing 5.04 kJ. Find its mass if kg·m2.

Energy and Momentum in Collisions

Bullet-Block System

Momentum and energy conservation are used to analyze collisions.

• Conservation of Momentum:

• Spring Compression:

• Example: A bullet embeds in a block, compressing a spring. Find the bullet's initial speed.

Energy Conservation with Nonconservative Forces

Work and Friction

Nonconservative forces like friction dissipate mechanical energy.

• Work-Energy Principle:

• Friction Work:

• Example: A 7.5 kg skier starts from rest, slides 65 m down a hill, friction does -1000 J of work. Find final speed.

Power

Rate of Doing Work

Power is the rate at which work is done or energy is transferred.

• Formula:

• Example: A power plant delivers 4.00 × 106 W over 5.0 s. Find the energy delivered.

Uniform Circular Motion

Speed and Centripetal Force

Objects moving in a circle at constant speed experience centripetal acceleration.

• Speed:

• Centripetal Acceleration:

• Example: A 1500-kg car moves at 25 m/s around a curve of diameter 200 m. Find acceleration.

Newton's Second Law in Dynamics

Force and Acceleration

Newton's Second Law relates force, mass, and acceleration.

• Formula:

• Example: A 6.0 kg block is pushed with 25 N. Find acceleration.

Relative Velocity

Reference Frames

Relative velocity is the velocity of an object as observed from a particular reference frame.

• Formula:

• Example: You drive at 65 mi/h east, pass a car going 50 mi/h west. Find your velocity relative to the other car.

Constant Acceleration Kinematics

Distance Traveled Under Acceleration

Kinematic equations describe motion under constant acceleration.

• Formula:

• Example: A jet accelerates from rest to 65 m/s in 3.0 s. Find distance traveled.

Vector Components and Addition

Resolving and Adding Vectors

Vectors can be broken into components and added using vector addition rules.

• Component Form: ,

• Resultant Vector:

• Example: Vector has magnitude 6.0, angle 30°, has magnitude 4.0, points west. Find .

Projectile Motion

Horizontal and Vertical Components

Projectile motion involves two-dimensional motion under gravity.

• Horizontal Range:

• Time of Flight:

• Example: A rock is thrown horizontally from a cliff at 22 m/s. Find time to hit water.

Collisions

Conservation of Momentum

In collisions, total momentum is conserved if no external forces act.

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Only momentum is conserved.

• Example: A 0.6 kg cart traveling at 3.0 m/s collides elastically with a 0.3 kg stationary cart. Find final speed of 0.3 kg cart.

Equilibrium and Tension

Forces in Hanging Objects

Objects in equilibrium have balanced forces and torques.

• Tension in Rope: For a mass hanging at an angle, resolve forces vertically and horizontally.

• Example: A 120 N mass hangs by a rope at 30° from the ceiling. Find tension in the rope.