Physics 21: Forces, Work, and Energy (Chapters 4–6) – Study Guide

Forces and Newton's Laws

Elevator Problems and Tension Forces

Elevator scenarios are classic applications of Newton's laws, especially the analysis of forces and tension in cables. Understanding these problems requires careful consideration of equilibrium and acceleration.

• Constant Speed (Up or Down): When an elevator moves at constant speed, it is in dynamic equilibrium. The upward force (tension) in the cable equals the downward gravitational force.

• Force of Gravity: $F_g = mg$

• Tension in Cable: For constant speed, $T = mg$; for acceleration $a$, $T = m(g + a)$ (upward) or $T = m(g - a)$ (downward).

• Work Done by Forces: Work is positive when the force and displacement are in the same direction, negative otherwise.

• Example: If an elevator is lifted at constant speed, the cable does positive work; if lowered, the cable does negative work.

Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. This law is fundamental in analyzing interactions between objects.

• Action-Reaction Pairs: The force exerted by a horse on a wagon is equal in magnitude and opposite in direction to the force exerted by the wagon on the horse.

• Application: In problems involving two interacting bodies, always identify the action-reaction pairs.

Work, Energy, and Power

Work-Kinetic Energy Theorem

The work-kinetic energy theorem relates the net work done on an object to its change in kinetic energy.

• Formula: $W_{net} = rac{1}{2}mv_f^2 - rac{1}{2}mv_i^2$

• Work by Multiple Forces: The total work is the sum of work done by all forces acting on the object.

• Example: A block sliding down an incline with friction: $W_{gravity} + W_{friction} = rac{1}{2}mv^2$

Work Done by Friction

Friction does negative work, removing energy from the system.

• Kinetic Friction Force: $f_k = bc_k N$

• Work by Friction: $W_{friction} = -f_k d$

• Variable Friction: If friction coefficient varies with distance, integrate: $W_{friction} = -int bc_k(x) N dx$

Work in Pulley Systems

Pulley systems often involve multiple masses and require careful accounting of work and energy transfer.

• Work on Each Block: Calculate work done on each block separately, considering direction and friction.

• Example: If a 20.0 N block moves 0.75 m to the right and a 12.0 N block moves 0.75 m downward, find work done on each using $W = F cdot d$.

Inclined Planes and Normal Forces

Normal Force on an Incline

The normal force is the perpendicular contact force exerted by a surface on an object.

• Formula: $N = mg cos alpha$ (where $alpha$ is the angle of the incline)

• Application: Used to calculate friction and analyze equilibrium on ramps.

Friction on Inclined Planes

Friction opposes motion and depends on the normal force and the coefficient of friction.

• Kinetic Friction: $f_k = mu_k N$

• Static Friction: $f_s leq mu_s N$

• Example: For a block on a ramp, calculate acceleration using $F_{net} = F_{applied} - f_k - mg sin alpha$

Spring Forces and Hooke's Law

Nonlinear Spring Force

Some springs do not obey Hooke's Law exactly and have force components that depend on higher powers of displacement.

• Generalized Spring Force: $F_x = kx + bx^2 + cx^3$

• Work to Stretch/Compress: $W = int F_x dx$

• Example: To stretch a spring by $x$, calculate $W = int_0^x (kx + bx^2 + cx^3) dx$

• Comparison: Nonlinear springs require more work to stretch/compress as $x$ increases.

Circular Motion and Tension

Tension in Rotating Systems

Objects moving in a circle experience centripetal force, which is often provided by tension in a string or cable.

• Centripetal Force: $F_c = frac{mv^2}{r}$

• Tension Calculation: For a rotating seat, resolve forces into components and use equilibrium conditions.

• Example: For a seat swinging at an angle, use trigonometry to find tension in each cable.

Equilibrium and Friction

Static Equilibrium

Systems in equilibrium have zero net force and zero net torque.

• Force Balance: $\sum F = 0$

• Friction Force: For a block at rest, $f_s = mu_s N$ up to its maximum value.

• Maximum Weight for Equilibrium: Set up equations for force components and solve for the maximum allowable weight.

Graphical Analysis of Motion

Velocity and Net Force Graphs

Graphs of velocity and net force versus time are useful for visualizing motion and force relationships.

• Constant Acceleration: If velocity increases linearly, net force is constant.

• Example: A graph showing $v_x$ increasing linearly implies $\sum F_x$ is constant and positive.

HTML Table: Comparison of Forces in Elevator Scenarios

Scenario	Tension in Cable	Work Done by Cable
Elevator moving up at constant speed	$T = mg$	Positive
Elevator moving down at constant speed	$T = mg$	Negative
Elevator accelerating upward	$T = m(g + a)$	Positive
Elevator accelerating downward	$T = m(g - a)$	Negative

Additional info:

• Some problems involve variable friction coefficients, requiring integration for work calculations.

• Nonlinear spring forces are less common but important for advanced applications.

• Graphical analysis is a key skill for interpreting motion and force relationships.