Formula Sheet: Core Concepts in Mechanics

Equations of Motion for Constant Acceleration

These equations describe the motion of objects under constant acceleration, fundamental in kinematics.

• Velocity as a function of time:

• Position as a function of time:

• Velocity-position relation:

• Displacement with average velocity:

Key Terms:

• Displacement (x - x0): Change in position.

• Initial velocity (v0x): Velocity at t = 0.

• Acceleration (ax): Rate of change of velocity.

Forces and Work

• Work by a constant force: Where F is force and s is displacement in the direction of force.

• Power: Rate at which work is done.

• Kinetic friction force: Where \( \mu_k \) is the coefficient of kinetic friction, n is the normal force.

• Static friction force: Where \( \mu_s \) is the coefficient of static friction.

• Spring force (Hooke's Law): Where k is the spring constant, x is displacement from equilibrium.

• Work done on a spring:

Momentum and Energy

• Momentum of a particle:

• Kinetic energy:

• Work-energy theorem: Total work equals change in kinetic energy.

Practice Test Topics and Applications

Contact Forces and Newton's Laws

Problems involving blocks in contact on frictionless surfaces test understanding of Newton's Third Law and force transmission.

• Example: Two blocks (6.00 kg and 4.00 kg) on a frictionless surface, with a 10.0 N force applied to the 6.00 kg block. Find the force exerted by the 6.00 kg block on the 4.00 kg block.

• Key Steps:

  • Calculate acceleration:

  • Find force on 4.00 kg block:

Friction and Stopping Distance

Frictional forces are crucial in determining stopping distances for vehicles.

• Stopping distance formula:

• Example: Car traveling at 28.7 m/s, (dry), (wet). Calculate stopping distance and required speed for same stopping distance on wet pavement.

Elevator Physics and Apparent Weight

Apparent weight changes with elevator acceleration, as measured by a scale.

• Apparent weight: (upward acceleration), (downward acceleration)

• Example: Scale readings for different elevator accelerations.

Tension in Cords and Systems of Masses

Analyzing tension in cords connecting masses under acceleration.

• Key Principle: Tension varies depending on the mass below each cord and the acceleration.

• Example: Four masses connected vertically, pulled upward. Calculate total pull and individual tensions.

Work Done by Gravity

Work done by gravity is the product of weight and vertical displacement.

• Formula:

• Example: Watermelon dropped from 18.0 m. Calculate work done by gravity.

Friction and Constant Speed

To maintain constant speed on a surface with friction, applied force must equal frictional force.

• Formula:

• Example: Pushing a sheep at constant speed, given mass and .

Force and Acceleration (Superman Problem)

Calculating the force required to accelerate an object horizontally.

• Formula:

• Example: Boulder of weight 1850 N, acceleration 11.3 m/s2.

Power and Energy Consumption

Energy used by electrical devices is the product of power and time.

• Formula:

• Example: 100-watt bulb used for 1 hour: J

Elastic Collisions

Elastic collisions conserve both momentum and kinetic energy.

• Conservation of momentum:

• Conservation of kinetic energy:

• Key Point: In perfectly elastic collisions, both total momentum and total kinetic energy are conserved.

Free-Body Diagrams and Elevator Motion

Free-body diagrams illustrate the forces acting on an object, such as a person in an accelerating elevator.

• Forces: (gravity, downward), (floor, upward)

• Key Principle: If elevator is slowing down while moving upward, net force is downward, so .

Summary Table: Key Formulas and Concepts

Additional info:

• Some questions reference diagrams and figures; in exam settings, always draw free-body diagrams to clarify force directions.

• For energy and power calculations, remember to convert units (e.g., hours to seconds).

• Stopping distance problems assume no other forces (e.g., air resistance) act except friction.