Fundamental Concepts in Newtonian Mechanics: Formula Sheet and Problem Applications

Formula Sheet for Exam 2

Trigonometric Relationships in Physics

Trigonometric functions are essential for resolving vector components, especially in problems involving inclined planes or forces at angles.

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Example: To find the vertical component of a force $F$ at angle $\theta$, use $F_y = F \sin \theta$.

Kinematic Equations for Constant Acceleration

Kinematic equations describe the motion of objects under constant acceleration, such as free-falling bodies or objects on inclined planes.

• $v_f = v_i + a \Delta t$

• $s_f = s_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$

• $v_f^2 = v_i^2 + 2a \Delta s$

• $\Delta t = t_f - t_i$

• $\Delta s = s_f - s_i$

• $\Delta v = v_f - v_i$

Example: Calculating the final velocity of a car accelerating from rest ($v_i = 0$) over a time interval $\Delta t$.

Newton's Second Law and Force Summation

Newton's Second Law relates the net force acting on an object to its acceleration and mass.

• $\sum \vec{F} = m \vec{a}$

• Acceleration due to gravity: $g = 9.8\ \text{m/s}^2$

Example: The net force required to accelerate a 5 kg object at $2\ \text{m/s}^2$ is $F = 5 \times 2 = 10\ \text{N}$.

Frictional Forces

Friction opposes the relative motion between surfaces. There are two main types: static and kinetic friction.

• Static friction: $F_f = \mu_s F_N$

• Kinetic friction: $F_f = \mu_k F_N$

Definitions:

• $\mu_s$ = coefficient of static friction

• $\mu_k$ = coefficient of kinetic friction

• $F_N$ = normal force (perpendicular to the contact surface)

Example: If $\mu_s = 0.5$ and $F_N = 20\ \text{N}$, then $F_f = 0.5 \times 20 = 10\ \text{N}$.

Exam 1: Problem Applications in Mechanics

Free-Body Diagrams

Free-body diagrams (FBDs) are graphical representations used to visualize the forces acting on an object. They are essential for solving problems involving equilibrium, tension, and friction.

• Key steps:

  1. Identify all forces acting on the object (gravity, normal force, tension, friction, applied forces).

  2. Represent each force as an arrow pointing in the direction of the force.

  3. Label each force clearly.

• Application: Used in problems involving suspended objects, boxes on slopes, and systems with pulleys.

Tension in Cables and Pulleys

Tension is the force transmitted through a string, cable, or rope when it is pulled tight by forces acting from opposite ends. In equilibrium, the sum of forces in each direction must be zero.

• For objects suspended by cables at angles:

  • Resolve tension forces into components using trigonometric functions.

  • Set up equilibrium equations for vertical and horizontal directions.

• For pulleys:

  • Assume massless and frictionless pulleys unless stated otherwise.

  • All masses connected by strings share the same magnitude of acceleration.

Example: For a traffic light suspended by two cables at different angles, use $\sum F_y = 0$ and $\sum F_x = 0$ to solve for tensions.

Forces in Elevators

When analyzing forces in elevators, consider the acceleration of the elevator and its effect on the normal force and tension in cables.

• Normal force on a box in an accelerating elevator:

  • If elevator accelerates upward: $F_N = m(g + a)$

  • If elevator accelerates downward: $F_N = m(g - a)$

• Tension in the cable:

  • Upward acceleration: $T = m(g + a)$

  • Downward acceleration: $T = m(g - a)$

Example: If a 10 kg box is in an elevator accelerating upward at $2\ \text{m/s}^2$, $F_N = 10(9.8 + 2) = 118\ \text{N}$.

Inclined Planes and Friction

Objects on inclined planes experience forces parallel and perpendicular to the surface. Friction determines whether the object remains stationary or slides down.

• Components of gravity:

  • Parallel: $mg \sin \theta$

  • Perpendicular: $mg \cos \theta$

• Static friction threshold: $mg \sin \theta_0 = \mu_s mg \cos \theta_0$

• Solving for critical angle: $\tan \theta_0 = \mu_s$

• Acceleration down the slope (kinetic friction): $a = g(\sin \theta - \mu_k \cos \theta)$

Example: For $\mu_s = 0.5$, the box starts to slide when $\theta_0 = \arctan(0.5) \approx 26.6^\circ$.

Systems of Connected Masses and Pulleys

Multiple masses connected by strings and pulleys require analysis of forces and acceleration for each mass. The system's acceleration is determined by the net force and total mass.

• Net force: $F_{\text{net}} = (m_1 + m_2 + m_3)a$

• Tension in strings: Analyze each mass separately using Newton's Second Law.

Example: For three masses connected over a pulley, set up equations for each mass and solve simultaneously for acceleration and tension.

Normal Force and Applied Forces

The normal force is the perpendicular contact force exerted by a surface on an object. Applied forces can cause objects to accelerate or overcome friction.

• Normal force on a horizontal surface: $F_N = mg$

• Normal force on an inclined plane: $F_N = mg \cos \theta$

• Maximum static friction: $F_{f,\text{max}} = \mu_s F_N$

• Applied force to move two blocks together: Set up equations to ensure both blocks accelerate together without slipping.

Example: If $m_1$ is on top of $m_2$ and $F$ is applied to $m_2$, the maximum $F$ before $m_1$ slips is $F_{\text{max}} = \mu_s m_1 g$.

Summary Table: Frictional Forces

Type of Friction	Formula	When Used
Static Friction	$F_f = \mu_s F_N$	Object at rest, prevents motion
Kinetic Friction	$F_f = \mu_k F_N$	Object in motion, opposes sliding

Summary Table: Kinematic Equations

Equation	Physical Meaning
$v_f = v_i + a \Delta t$	Final velocity after time $\Delta t$
$s_f = s_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$	Final position after time $\Delta t$
$v_f^2 = v_i^2 + 2a \Delta s$	Relates velocity and displacement

Additional info: These notes expand upon the formula sheet and exam questions by providing definitions, examples, and context for each concept, ensuring a comprehensive review for students preparing for introductory college-level physics exams.