Physics I: Mechanics – Friction, Ramps, Pulleys, and Circular Motion

Friction

Introduction to Friction

Friction is a resistive force that acts between two surfaces in contact, opposing their relative motion. It plays a crucial role in everyday phenomena and engineering applications.

• Static Friction (f_s): The force that prevents relative motion between surfaces at rest. It increases with applied force up to a maximum value.

• Kinetic Friction (f_k): The force that opposes motion once surfaces are sliding past each other. It is generally constant and less than the maximum static friction.

• Normal Force (n): The perpendicular contact force exerted by a surface.

Key Equations:

• Maximum static friction: $f_s^{max} = oldsymbol{ ext{μ}_s n}$

• Kinetic friction: $f_k = oldsymbol{ ext{μ}_k n}$

Example:

If a box is at rest, static friction equals the applied force until the maximum is reached. Once the box moves, kinetic friction takes over and remains nearly constant.

Sample Problem: Pulling a Block

To move a block along a rough surface, a force is applied at an angle above the horizontal. Given coefficients of friction ($ext{μ}_s = 0.9$, $ext{μ}_k = 0.4$), determine the minimum force required to initiate motion.

• Resolve the applied force into horizontal and vertical components.

• Calculate the normal force, accounting for the vertical component of the applied force.

• Set the horizontal component equal to the maximum static friction to solve for the required force.

Equation:

• $F_{min} ext{ (horizontal)} = ext{μ}_s n$

Example:

Pulled at $42^ ext{o}$ above horizontal, the force needed is found by balancing the horizontal component with the maximum static friction.

Atwood’s Machine

System Description

An Atwood’s Machine consists of two masses connected by a rope over a frictionless pulley. It is a classic system for studying Newton’s laws and acceleration due to gravity.

• Free-body diagrams: Draw for each mass, showing tension and gravitational forces.

• Assumptions: Pulley is frictionless, rope is massless.

Solving for Acceleration and Tension

To find the acceleration and tension in the rope, apply Newton’s second law to each mass and solve the system of equations.

Equations:

• For mass $m_1$ (moving up): $T - m_1g = m_1a$

• For mass $m_2$ (moving down): $m_2g - T = m_2a$

• Combine to solve for acceleration: $a = rac{g(m_2 - m_1)}{m_1 + m_2}$

• Tension: $T = m_1a + m_1g$

Example:

Given $m_1 =$ mass of bricks, $m_2 = 28.0$ kg, calculate $a$ and $T$ using the above formulas.

Inclined Plane

Forces on an Inclined Plane

When an object rests or moves on an inclined plane, its weight is resolved into components parallel and perpendicular to the surface.

• Parallel component: $w ext{sin} heta$ (causes sliding down the plane)

• Perpendicular component: $w ext{cos} heta$ (normal force)

Diagram:

Vectors show the decomposition of weight into $w ext{sin} heta$ and $w ext{cos} heta$.

Sled Down an Inclined Plane

When sliding down, the net force is the parallel component minus friction (if present).

• Without friction: $a = g ext{sin} heta$

• With friction: $a = g( ext{sin} heta - ext{μ}_k ext{cos} heta)$

Inclined Plane with Friction

Friction opposes the motion, reducing acceleration. The kinetic friction force is $f_k = ext{μ}_k n$.

• Net force: $F_{net} = m g ext{sin} heta - f_k$

• Normal force: $n = m g ext{cos} heta$

Inclined Plane + Pulley

Combining an inclined plane with a pulley system allows analysis of more complex motion, such as two masses connected over a pulley, one on the plane and one hanging.

• Set up force equations for each mass.

• Include friction for the mass on the plane.

• Solve for acceleration using Newton’s second law.

Equation:

• $a = rac{m_2g - m_1g ext{sin} heta - ext{μ}_k m_1g ext{cos} heta}{m_1 + m_2}$

Example:

Given $m_1 = 100$ kg, $m_2$ unknown, $ext{μ}_k = 0.4$, find the acceleration.

Sample Problem: Maximum Acceleration Up a Ramp

Given a 5,000 kg pallet, ramp angle $60^ ext{o}$, cable breaking tension 70,000 N, and $ext{μ}_k = 0.95$, find the maximum acceleration before the cable breaks.

• Sum forces along the ramp: $T - m g ext{sin} heta - f_k = m a$

• Maximum tension: $T_{max} = 70,000$ N

• Friction: $f_k = ext{μ}_k m g ext{cos} heta$

• Solve for $a$:

$a = rac{T_{max} - m g ext{sin} heta - ext{μ}_k m g ext{cos} heta}{m}$ Example:

Plug in values to find $a$.

Circular Motion

Uniform Circular Motion

Uniform circular motion occurs when an object moves at constant speed along a circular path. The velocity vector changes direction, resulting in acceleration toward the center of the circle (centripetal acceleration).

• Centripetal acceleration: Always points toward the center.

• No parallel component: Acceleration is perpendicular to velocity.

Equation:

• $a_c = rac{v^2}{R}$

Period and Frequency:

• Time for one revolution (period): $T = rac{2 ext{π}R}{v}$

Example:

A particle moving at constant speed $v$ around a circle of radius $R$ experiences centripetal acceleration $a_c$.

Summary Table: Friction Types

Type	Symbol	Equation	When Used
Static Friction	$f_s$	$f_s ext{≤} ext{μ}_s n$	Object at rest
Kinetic Friction	$f_k$	$f_k = ext{μ}_k n$	Object in motion

Summary Table: Forces on Inclined Plane

Component	Equation	Description
Parallel	$w ext{sin} heta$	Down the plane
Perpendicular	$w ext{cos} heta$	Normal to the plane

Summary Table: Atwood’s Machine Equations

Quantity	Equation
Acceleration	$a = rac{g(m_2 - m_1)}{m_1 + m_2}$
Tension	$T = m_1a + m_1g$

Summary Table: Circular Motion

Quantity	Equation
Centripetal Acceleration	$a_c = rac{v^2}{R}$
Period	$T = rac{2 ext{π}R}{v}$

Additional info: Some diagrams and equations have been expanded for clarity and completeness. All equations are presented in standard LaTeX format for academic use.