Physics for Life Sciences I: Forces, Friction, Drag, and Interacting Objects (Lecture 9, Sections 5.5-5.8)

General Strategy for Using Newton's Second Law

Newton's Second Law in Component Form

Newton's second law relates the net force acting on an object to its acceleration. In component form, it is written as:

• Equation:

$\sum F_x = m a_x$ $\sum F_y = m a_y$

• Equilibrium Problems: If an object is in equilibrium (either static, $\vec{v}=0$, or dynamic, $\vec{v}\neq0$ but constant), its acceleration components are zero: $a_x = a_y = 0$.

• Dynamics Problems: If the object is accelerating, either the forces are known and acceleration is sought, or vice versa.

Apparent Weight

Definition and Calculation

Apparent weight is the magnitude of the contact force supporting an object, such as what a scale reads. It is your sensation of weight and can differ from your true weight depending on acceleration.

• True Weight: $w = m g$

• Apparent Weight (when vertical acceleration is present):

$\sum F_y = n_y + w_y = n - w = m a_y$ $w_{app} = w + m a_y$

• When $a_y = 0$, $w_{app} = w$.

• When accelerating upward, $w_{app} > w$; downward, $w_{app} < w$.

Normal Forces

Definition and Properties

The normal force is the force preventing an object from penetrating a surface. It is always perpendicular to the surface and directed away from it.

• On an inclined plane, the normal force points perpendicular to the surface.

• Magnitude:

$n - w \cdot \cos(\theta) = m a_y$

• Use Newton's second law along the direction perpendicular to the surface to solve for $n$.

Section 5.5: Friction

Static Friction

Static friction is the force a surface exerts to keep an object from sliding. It opposes the direction in which the object would move if there were no friction.

• Direction: Opposes potential motion.

• Magnitude: Adjusts to balance applied force, up to a maximum value.

• Maximum Static Friction:

$f_{s, max} = \mu_s n$

• $\mu_s$ is the coefficient of static friction.

• If the required friction exceeds $f_{s, max}$, the object will slip and start moving.

Kinetic Friction

Kinetic friction acts when an object is sliding. Its magnitude is nearly constant and given by:

$f_k = \mu_k n$

• $\mu_k$ is the coefficient of kinetic friction.

• Direction: Opposite to the direction of motion.

• Kinetic friction does not depend on speed.

Rolling Friction

Rolling friction opposes the motion of a rolling object. It is generally much smaller than static or kinetic friction.

$f_r = \mu_r n$

• $\mu_r$ is the coefficient of rolling friction.

Coefficients of Friction (Table)

The coefficients of friction depend on the materials in contact. Typical values are:

Materials	Static ($\mu_s$)	Kinetic ($\mu_k$)	Rolling ($\mu_r$)
Steel on steel	0.80	0.60	0.002
Ice on steel	0.10	0.05	—
Wood on wood	0.50	0.20	—
Rubber on concrete	1.0	0.8	—
Ice on ice	0.10	0.03	—
Additional info: Rolling friction coefficients are typically much smaller than static or kinetic friction.

Working with Friction Forces

• If the object is not moving: Use static friction, $f_s$ (magnitude $\leq \mu_s n$).

• If the object is sliding: Use kinetic friction, $f_k = \mu_k n$.

• If the object is rolling: Use rolling friction, $f_r = \mu_r n$.

Example: Finding the Force to Slide a Sofa

To move a 32 kg sofa with sliders ($\mu_k = 0.08$) at constant speed:

• Normal force: $n = mg$

• Kinetic friction: $f_k = \mu_k n = \mu_k mg$

• Required force: $F = f_k = (0.08)(32)(9.8) = 25$ N

• Speed does not affect the required force.

Section 5.6: Drag

Drag Force

Drag is a force that opposes the motion of an object through a fluid (such as air or water). It increases with speed and depends on fluid properties.

• Direction: Opposite to velocity.

• Magnitude: Increases with speed.

Reynolds Number

The Reynolds number ($Re$) characterizes the type of drag:

$Re = \frac{\text{inertial forces}}{\text{viscous forces}} = \frac{\rho v L}{\eta}$

• High $Re$ ($>1000$): Inertial forces dominate (e.g., baseball in air).

• Low $Re$ ($<1$): Viscous forces dominate (e.g., bacteria in water).

Drag at High Reynolds Number

For most large objects moving quickly through air:

$D = C_p \rho A v^2$

• $C_p$ is the drag coefficient (depends on shape).

• $A$ is cross-sectional area.

• Drag force is proportional to $v^2$.

Drag at Low Reynolds Number (Stokes' Law)

For small objects moving slowly in viscous fluids:

$D = 6 \pi \eta r v$

• $\eta$ is fluid viscosity.

• $r$ is object radius.

• Drag force is proportional to $v$ (linear drag).

Terminal Speed

When drag force balances the applied force (e.g., gravity), the object reaches a constant terminal speed:

• At terminal speed, $D = F_{applied}$ and $a = 0$.

• For falling objects, terminal speed is reached when drag equals weight.

Section 5.7: Interacting Objects

Newton's Third Law

Newton's third law states that every force occurs as one member of an action/reaction pair, acting on different objects, equal in magnitude and opposite in direction.

• Action/reaction pairs: $F_{A\rightarrow B} = -F_{B\rightarrow A}$

• For objects in contact, their accelerations are linked.

Analyzing Objects in Contact

• Draw separate force identification and free-body diagrams for each object.

• Write Newton's second law for each object.

• Action/reaction pairs have equal magnitude, opposite direction.

• Objects in contact have the same acceleration.

Example: Pushing Two Blocks

Two blocks (A: 5 kg, B: 10 kg) are pushed together with a force. The force exerted by A on B is found by:

• Write Newton's second law for each block.

• Set accelerations equal ($a_A = a_B$).

• Solve simultaneous equations to find contact force.

Section 5.8: Ropes and Pulleys

Tension in Ropes

In a massless rope or string, the tension is the same throughout and equals the magnitude of the force pulling on the rope.

• Massless rope transmits force undiminished from one end to the other.

• Tension is the same on both sides of a massless, frictionless pulley.

Working with Ropes and Pulleys

• Draw free-body diagrams for each object.

• Write Newton's second law for each object.

• For connected objects, their accelerations are related (often equal in magnitude, opposite in direction).

Example: Lifting a Stage Set

A 200 kg set and a 100 kg stagehand are connected by a rope over a pulley. When the rope is released, both accelerate. The acceleration is found by:

• Write Newton's second law for each object:

$T - m_{man}g = m_{man} a_{man}$ $T - m_{set}g = -m_{set} a_{man}$

• Set $a_{set} = -a_{man}$ and solve for $a_{man}$:

$a_{man} = \frac{m_{set} - m_{man}}{m_{set} + m_{man}} g$

• Plug in values to find acceleration.

Summary of Important Forces

• Weight: $w = mg$ (downward)

• Static friction: $f_s$ (up to $\mu_s n$, direction opposes motion)

• Kinetic friction: $f_k = \mu_k n$ (direction opposite motion)

• Rolling friction: $f_r = \mu_r n$

• Drag: $D$ (opposite velocity, $C_p \rho A v^2$ or $6\pi \eta r v$)

• Tension: Same throughout a massless rope or string, unchanged by massless, frictionless pulleys.

Applications

• Terminal speed: Object moves at constant speed when drag balances applied force.

• Ropes and pulleys: Tension is equal throughout, useful for analyzing connected systems.

Additional info: These notes cover the fundamental forces and interactions relevant to introductory physics, with emphasis on friction, drag, and systems of interacting objects. Examples and equations are provided for practical problem-solving.