Newton's Third Law and Connected Systems: Study Guide

Newton's Third Law

Concept and Definition

Newton's Third Law of Motion is a fundamental principle in classical mechanics, stating that for every action, there is an equal and opposite reaction. This law governs the interactions between objects and is essential for understanding force pairs in physical systems.

• Newton's Third Law: For every action (force), there is a reaction of equal magnitude but opposite direction.

• Action-Reaction Force Pairs: All forces exist in pairs, acting on two different objects.

• Key Point: Action-reaction pairs do not imply equal accelerations; acceleration depends on mass.

• Formula: $\Sigma F = ma$ (Sum of forces equals mass times acceleration)

Example: If you push a 40 kg ice block with a force of 20 N, the block exerts a force of 20 N back on you, but the accelerations differ due to different masses.

Identifying Action-Reaction Pairs

Not all force pairs are action-reaction pairs. For example, the weight of a book and the normal force on the book are not an action-reaction pair; instead, the weight of the book from the Earth and the gravitational force on the Earth from the book are.

• Action-Reaction Pair: Friction on the book from the floor & friction on the floor from the book.

• Not an Action-Reaction Pair: Weight of the book & normal force on the book.

Force Problems in Connected Systems

Connected Objects (X-Axis)

When objects are connected (e.g., by strings or cables), they move together with the same acceleration and velocity. Analyzing such systems requires careful application of Newton's laws.

• Draw Free-Body Diagrams (FBDs): Identify all forces acting on each object.

• Write Equations: $\Sigma F = ma$ for each object.

• Solving Methods: Use equation addition or substitution to eliminate non-target variables.

Example: Two blocks (3 kg and 5 kg) connected by a string, pulled with 30 N. Find acceleration and tension.

Systems of Objects with Multiple Strings

When objects hang by multiple ropes or strings, each tension supports the total weight below it.

• Draw FBDs: For each object.

• Write $\Sigma F = ma$: Start with the simplest object.

• Solve for Tension: The top-most string supports the total weight below.

Systems with Pulleys

In systems with massless pulleys, the tension in the string is the same on both sides, but the direction differs. The acceleration is shared by all connected objects.

• Direction of Positive: Usually the direction the heavier object will fall.

• Draw FBDs: For all objects.

• Write $\Sigma F = ma$: Start with the simplest object.

Example: Atwood Machine: Two blocks (6 kg and 4 kg) connected by a string over a pulley. Find acceleration and tension.

Combining Systems into a Single Object

To simplify calculations, combine all masses into a single object and ignore internal forces (tensions, normals) between them.

• Shortcut: $M = m_A + m_B + ...$

• Ignore Internal Forces: Only consider external forces.

Example: Two blocks (3 kg and 5 kg) pulled with 30 N. Combine masses to find system acceleration.

Connected Systems with Friction

Friction in Connected Systems

When friction is present, it must be considered for each object. Connected objects still share the same acceleration and velocity.

• Types of Friction: Static ($\mu_s$) and kinetic ($\mu_k$).

• Determine Friction: If the sum of forces exceeds maximum static friction, use kinetic friction.

• Formula: $f_s \leq \mu_s N$, $f_k = \mu_k N$

Example: 10 kg and 5 kg blocks on a rough table ($\mu_s = 0.5$, $\mu_k = 0.3$), pulled with 90 N. Find acceleration.

Minimum Mass to Prevent Motion

To prevent motion, the static friction must be sufficient to counteract the force from the hanging block.

• Calculate $m_A$: Set $f_s = \mu_s m_A g$ equal to the force from the hanging block.

Inclined Planes and Ramps

Solving Inclined Plane Problems

On inclined planes, the axes are tilted to align with the slope. The gravitational force must be decomposed into components parallel and perpendicular to the incline.

• Decompose $mg$: $mg_x = mg \sin \theta$, $mg_y = mg \cos \theta$

• Acceleration: Only occurs along the x-axis; $a_y = 0$

• Normal Force: $N = mg \cos \theta$

Example: 5 kg block on a frictionless incline at 37°. Find acceleration and normal force.

Inclined Planes with Friction

When friction is present, compare the net force along the axis of motion to the maximum static friction to determine if the object moves.

• Static Friction: $f_s \leq \mu_s N$

• Kinetic Friction: $f_k = \mu_k N$

• Net Force: $\Sigma F = ma$

Example: 10 kg block on a 37° ramp ($\mu_s = 0.6$, $\mu_k = 0.4$). Find friction force and acceleration.

Critical Angles on Rough Inclined Planes

Critical angles are the angles at which an object just begins to slide (static) or slides at constant speed (kinetic). These depend only on the coefficient of friction.

• Static Critical Angle: $\theta_{crit,s} = \tan^{-1}(\mu_s)$

• Kinetic Critical Angle: $\theta_{crit,k} = \tan^{-1}(\mu_k)$

• Coefficient from Angle: $\mu_s = \tan(\theta_{crit,s})$, $\mu_k = \tan(\theta_{crit,k})$

Example: 6 kg block on ramp, $\mu_s = 0.75$. Find $\theta_{crit,s}$.

Connected Objects on Inclined Planes with Friction

Combining Multiple Concepts

Problems may involve multiple objects on ramps with friction, requiring the use of all previous concepts.

• Draw FBDs: For all objects, tilt axes as needed.

• Determine Friction: Use $\mu_s$ or $\mu_k$ as appropriate.

• Write $\Sigma F = ma$: Start with the simplest object.

• Solve for Acceleration: Use equation addition or substitution.

Example: Two blocks connected by a cable and pulley, one moving up a 30° incline with $\mu_k = 0.15$. Find acceleration.

Stacked Blocks and Friction

Stacked Objects

When objects are stacked, the friction between them determines whether they move together or slide relative to each other.

• Friction Direction: Acts in the same direction as motion.

• Static Friction: When relative velocity is zero.

• Kinetic Friction: When relative velocity is not zero.

• Maximum Acceleration: $a_{max} = \mu_s g$ (for top block to not slide)

Example: 10 kg box with 5 kg box on top ($\mu_s = 0.7$, $\mu_k = 0.3$). Find maximum acceleration for both to move together.

Problems with Stacked Blocks

Calculate tension or maximum force to keep stacked objects moving together, considering friction between surfaces.

• Draw FBDs: For all objects.

• Determine Friction: Use $\mu_s$ or $\mu_k$ as appropriate.

• Write $\Sigma F = ma$: Start with the simplest object.

Example: 5 kg block A on 10 kg block B, A tied to wall, B pulled with 45 N ($\mu_k = 0.2$). Find tension on A.

Summary Table: Friction and Inclined Planes

Type	Formula	Key Variable
Static Friction	$f_s \leq \mu_s N$	$\mu_s$
Kinetic Friction	$f_k = \mu_k N$	$\mu_k$
Inclined Plane (mg components)	$mg_x = mg \sin \theta$, $mg_y = mg \cos \theta$	$\theta$
Critical Angle (static)	$\theta_{crit,s} = \tan^{-1}(\mu_s)$	$\mu_s$
Critical Angle (kinetic)	$\theta_{crit,k} = \tan^{-1}(\mu_k)$	$\mu_k$