circular motion lec1

Introduction to Circular Motion and Basic Kinematics Review

• Concept of Circular Motion: Circular motion occurs when a body moves in a circular path or a "dayera."

• Review of Linear Quantities:

  • Distance: The total path covered by a body.

  • Displacement: The straight-line distance from the initial to the final point.

  • Velocity: Displacement per unit time ($v = \frac{d}{t}$).

  • Acceleration: The rate of change of velocity ($a = \frac{\Delta v}{t}$).

• Transition to Angular Quantities: In circular motion, we analyze angular counterparts to linear quantities:

  • Linear Displacement $s$ → Angular Displacement $\theta$

  • Linear Velocity $v$ → Angular Velocity $\omega$

  • Linear Acceleration $a$ → Angular Acceleration $\alpha$

Angular Displacement and Arc Length

• Definition: Angular displacement ($\theta$) is the angle subtended at the center of the circular path during circular motion.

• Position Vector ($\mathbf{r}$):

  • It defines the location or address of a point relative to the origin.

  • In circular motion, the magnitude of the position vector is the radius ($r$).

  • Example: A vector $5i$ meters means a point located 5 meters along the positive x-axis from the origin.

  • Standard unit vectors are $i$ (x-axis), $j$ (y-axis), and $k$ (z-axis).

• Arc Length ($s$):

  • The distance covered along the circumference (boundary) of the circle.

  • Relationship: $s = r\theta$.

  • In this formula, $\theta$ must be in radians.

• Vector Nature of $\theta$:

  • Angular displacement is considered a vector quantity only for very small values (infinitesimal values).

  • This is shown using the limit notation: $\lim_{\Delta t \to 0}$.

  • Small arc lengths approximate a straight line, satisfying the vector definition of linear displacement, which makes associated angular displacement a vector.

Units and Conversions

• Common Unit: Degree ($^{\circ}$).

• Standard (SI) Unit: Radian (rad).

• Total Cycle Relations:

  • One complete revolution (~360 degrees) is equal to $2\pi$ radians.

  • $2\pi \, \text{rad} = 360^{\circ}$

  • $\pi \, \text{rad} = 180^{\circ}$

• Specific Conversion Values:

  • $1 \, \text{rad} = \frac{180}{\pi} \approx 57.3^{\circ}$

  • $1^{\circ} = \frac{\pi}{180} \approx 0.01745 \, \text{rad}$

• Conversion Procedure:

  • To convert Degrees to Radians: Multiply by $\frac{\pi}{180}$.

  • To convert Radians to Degrees: Multiply by $\frac{180}{\pi}$.

• Common Angles:

  • $90^{\circ} = \frac{\pi}{2} \, \text{rad}$

  • $45^{\circ} = \frac{\pi}{4} \, \text{rad}$

  • $60^{\circ} = \frac{\pi}{3} \, \text{rad}$

Angular Velocity ($\omega$)

• Definition: The rate of change of angular displacement ($\omega = \frac{\theta}{t}$).

• Standard Unit: $\text{rad/s}$.

• Common Units: RPM (Rotation Per Minute).

  • To convert 1 RPM to rad/s: Multiply by $\frac{2\pi}{60}$ or $\frac{\pi}{30}$.

• Time Period ($T$): The time required to complete one rotation. For a complete cycle, $\omega = \frac{2\pi}{T}$.

• Direction (Right-Hand Rule):

  • Curl the fingers of the right hand along the direction of rotation.

  • The extended thumb points in the direction of $\omega$.

  • Anti-Clockwise (Counter-Clockwise): Direction is Outward from the screen.

  • Clockwise: Direction is Inward to the screen.

• Axial Vector: $\omega$ is an axial vector because its direction always lies along the axis of rotation, regardless of the clockwise or anti-clockwise sense.

Dynamics of Clock Hands (Angular Speed Examples)

• Second Hand:

  • Time Period ($T$) = $60 \, \text{s}$.

  • $\omega = \frac{2\pi}{60} = \frac{\pi}{30} \, \text{rad/s}$.

• Minute Hand:

  • Time Period ($T$) = $1 \, \text{hour} = 3600 \, \text{s}$.

  • $\omega = \frac{2\pi}{3600} = \frac{\pi}{1800} \, \text{rad/s}$.

• Hour Hand:

  • Time Period ($T$) = $12 \, \text{hours} = 12 \times 3600 = 43,200 \, \text{s}$.

  • $\omega = \frac{2\pi}{43,200} = \frac{\pi}{21,600} \, \text{rad/s}$.

Angular Acceleration ($\alpha$)

• Definition: The rate of change of angular velocity ($\alpha = \frac{\Delta \omega}{t}$).

• Unit: $\text{rad/s}^2$.

• Vector Nature: Like $\omega$, $\alpha$ is directed along the axis of rotation.

• Directionality:

  • If total angular speed is increasing ($\Delta \omega$ is positive): $\alpha$ is parallel to $\omega$.

  • If total angular speed is decreasing ($\Delta \omega$ is negative/retardation): $\alpha$ is anti-parallel to $\omega$.

  • Example: When a fan is switched on, $\alpha$ and $\omega$ are parallel. When switched off, they are anti-parallel.

  • If $\omega$ is constant, $\alpha = 0$.

Linear and Angular Relationships

• Velocity: $v = r\omega$ (Vector form: $\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}$). Note: $\mathbf{r} \times \boldsymbol{\omega}$ is incorrect due to the non-commutative property of cross products.

• Acceleration: $a_t = r\alpha$ (Vector form: $\mathbf{a_t} = \boldsymbol{\alpha} \times \mathbf{r}$).

• Equations of Motion for Constant $\alpha$:

  • $\omega_f = \omega_i + \alpha t$

  • $\theta = \omega_i t + \frac{1}{2} \alpha t^2$

  • $2\alpha\theta = \omega_f^2 - \omega_i^2$

• Total Revolutions: $\theta_{total} = 2\pi n$, where $n$ is the number of rotations.

Centripetal and Tangential Acceleration

• Tangential Acceleration ($a_t$):

  • Related to changes in the magnitude (speed) of tangential velocity.

  • Acts along the tangent.

• Centripetal (Radial) Acceleration ($a_c$):

  • Related to changes in the direction of velocity.

  • Always points towards the center of the circle.

  • Formula: $a_c = \frac{v^2}{r} = r\omega^2$.

• Net Linear Acceleration:

  • $a_{net} = \sqrt{a_c^2 + a_t^2}$.

• Angular relationships with the Radius Vector (r):

  • The angle between the radius vector ($r$) and centripetal acceleration ($a_c$) is $180^{\circ}$ ($\pi$ radians).

  • The angle between $a_c$ and $a_t$ is $90^{\circ}$ ($\frac{\pi}{2}$ radians).

Uniform Circular Motion (UCM)

• Definition: Circular motion with a constant speed.

• Characteristics:

  • $a_t = 0$ and $\alpha = 0$.

  • $a_c \neq 0$ because direction changes constantly.

  • Velocity is not constant (direction changes).

  • Acceleration is variable because its direction changes, even if the magnitude ($a_c = \frac{v^2}{r}$) remains constant.

Horizontal and Vertical Circular Motion

• Horizontal Circular Motion:

  • Force of gravity acts downward, not affecting the horizontal path.

  • Centripetal force ($F_c$) is provided entirely by the Tension ($T$) in the string.

  • $T = \frac{mv^2}{r}$.

• Vertical Circular Motion (Non-Uniform):

  • Gravity significantly affects speed (acceleration downward, deceleration upward).

  • At the Bottom (Lowest Point):

    • Tension and Weight are opposite.

    • $T_{max} = F_c + w = \frac{mv^2}{r} + mg$.

  • At the Top (Highest Point):

    • Tension and Weight both act downward.

    • $T_{min} = F_c - w = \frac{mv^2}{r} - mg$.

• Critical Velocities:

  • To maintain a circle without the string slackening, minimum velocity at the top: $v_{top} = \sqrt{gr}$.

  • Minimum velocity required at the bottom to complete the circle: $v_{bottom} = \sqrt{5gr}$.

Questions & Discussion

• Q: Can a body move in UCM with constant velocity?

  • A: No. Velocity is a vector. Even if speed is constant, the direction changes at every point on the circle, thus velocity is variable.

• Q: Where is the string most likely to break in vertical motion?

  • A: At the bottom, where tension is maximum ($T = F_c + mg$).

• Q: What provides the centripetal force for a planet?

  • A: Gravitational force.

• Q: What is the advice for students regarding past papers?

  • A: While past papers (UET, Mehran, NED, Nums) often repeat questions, relying solely on them is a gamble. Full course coverage is the first priority. If using past papers as a last resort, solve all available years, not just the most recent ones.

• Q: How is 2nd-year physics being handled?

  • A: There are three options: attend the concurrent teacher's live sessions, watch the lecturer's own recorded sessions from earlier batches, or wait for the lecturer to cover it personally after May 20th.