Force and Equilibrium

Torque and Rotational Equilibrium

Rotational equilibrium occurs when the sum of all torques acting on a system is zero, resulting in no angular acceleration. This principle is often applied to problems involving doors, beams, and other rotating bodies.

• Torque (τ): The rotational analog of force, defined as the product of force and the perpendicular distance from the axis of rotation.

• Equation:

• Equilibrium Condition:

• Example: Two children push on opposite sides of a door. If one pushes with at and the other at , the force needed for equilibrium is:

• Application: Used to determine forces required to keep doors, beams, or levers stationary.

Springs and Hooke's Law

Hooke's Law describes the force exerted by a spring when it is stretched or compressed.

• Hooke's Law:

• Spring Constant (k): Measures the stiffness of the spring (units: N/m).

• Direction: The force exerted by the spring is always directed toward the equilibrium position.

• Example: For and , (toward equilibrium).

Statics: Beams and Friction

Uniform Beam Supported by Cable

When analyzing beams supported by cables and walls, both torque and friction must be considered to prevent slipping.

• Static Friction: where is the coefficient of static friction and is the normal force.

• Torque Balance: Used to determine the minimum distance from the support point to prevent slipping.

• Example: For a beam of length , , and weight , calculate the minimum distance from point A for equilibrium.

Impulse, Momentum, and Energy

Impulse and Change in Momentum

Impulse is the product of force and the time interval over which it acts, resulting in a change in momentum.

• Impulse (J):

• Momentum (p):

• Example: A car is hit with for :

• Direction: The direction of impulse matches the direction of the applied force.

Conservation of Momentum in Collisions

In the absence of external forces, the total momentum of a system remains constant during a collision.

• Conservation Law: (for perfectly inelastic collisions)

• Types of Collisions:

  • Elastic: Both momentum and kinetic energy are conserved.

  • Inelastic: Only momentum is conserved; kinetic energy is not.

  • Perfectly Inelastic: Objects stick together after collision.

• Example: A 1000 kg car moving east at 20 m/s collides with a 1500 kg van moving north at 30 m/s. If they stick together, use vector addition to find the final velocity.

Recoil Velocity (Conservation of Momentum)

When an object (e.g., a cannon) fires a projectile, the recoil velocity can be found using conservation of momentum.

• Equation:

• Example: A 2650 kg cannon fires a 16.0 kg shell at 480 m/s at 20° above horizontal. The horizontal component of shell velocity is .

• Recoil velocity:

Momentum, Collisions, Rotations

Ballistic Pendulum and Conservation Laws

A ballistic pendulum is used to measure the speed of a projectile by analyzing the collision and subsequent motion.

• Perfectly Inelastic Collision: The bullet embeds in the block; both move together after impact.

• Conservation of Linear Momentum:

• Conservation of Energy (after collision): The combined mass swings upward, converting kinetic energy to potential energy.

• Maximum Vertical Displacement: Use energy conservation:

Springs and Friction

When a mass is pressed against a spring and released, it can slide across a surface, losing energy to friction.

• Work Done by Spring:

• Work Done by Friction:

• Maximum Displacement: Set spring work equal to friction work to solve for .

Rotational Motion and Angular Momentum

Angular momentum is conserved in the absence of external torques. This principle is used in problems involving rotating doors and objects.

• Angular Momentum (L): where is the moment of inertia and is angular velocity.

• Torque:

• Example: Throwing a ball at a door to just barely close it, considering the door's moment of inertia and frictional torque.

Summary Table: Types of Collisions

Additional info: These notes expand on the original questions by providing definitions, formulas, and context for each topic, ensuring a self-contained study guide for exam preparation.