Physics for Life Sciences I: Circular Motion, Gravity, and Rotational Motion (Lecture 11 Study Notes)

Circular Motion and Its Description

Describing Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle at a constant speed. Key quantities used to describe this motion are period, frequency, and speed.

• Period (T): The time required for one complete revolution around the circle.

• Frequency (f): The number of revolutions per second. It is the reciprocal of the period: $f = \frac{1}{T}$

• Speed (v): The linear speed along the circle, related to period and radius by: $v = 2\pi r f = \frac{2\pi r}{T}$

Example: If a particle completes 5 revolutions per second on a circle of radius 2 m, its speed is $v = 2\pi \times 2 \times 5 = 20\pi$ m/s.

Forces in Uniform Circular Motion

Centripetal Acceleration and Force

Even if the speed is constant, the direction of velocity changes continuously in circular motion, resulting in acceleration toward the center (centripetal acceleration).

• Centripetal Acceleration (a): $a = \frac{v^2}{r}$, always directed toward the center of the circle.

• Net Force (F): Required to maintain circular motion, also directed toward the center: $F = m a = m \frac{v^2}{r}$

Example: A 1 kg mass moving at 2 m/s in a circle of radius 1 m requires a net force of $F = 1 \times \frac{2^2}{1} = 4$ N toward the center.

Orbital Motion and Apparent Weight

Orbital Motion

An object can maintain a stable orbit if the gravitational force provides the necessary centripetal force.

• Orbital Speed: $v_{\text{orbit}} = \sqrt{g r}$, where $g$ is the gravitational acceleration and $r$ is the orbital radius.

Apparent Weight and Weightlessness

Apparent weight is the normal force experienced by an object, which may differ from true weight in non-inertial frames or during circular motion.

• Apparent Weight ($W_{\text{app}}$): The normal force $n$ exerted by a surface. $W_{\text{app}} = n$.

• In orbital motion, both astronaut and spacecraft are in free fall, so $n = 0$ and the astronaut feels weightless.

Newton's Law of Universal Gravitation

Gravity as an Inverse-Square Law

Gravity is a universal force acting between all masses. Newton's law states:

• The force is inversely proportional to the square of the distance between objects.

• The force is directly proportional to the product of the two masses.

Newton's Law of Gravity:

• For two masses $m_1$ and $m_2$ separated by distance $r$: $F = G \frac{m_1 m_2}{r^2}$ where $G = 6.67 \times 10^{-11}$ N·m2/kg2 is the gravitational constant.

Example: The gravitational force between two 65 kg students sitting 0.60 m apart is: $F = 6.67 \times 10^{-11} \frac{65 \times 65}{0.60^2} \approx 7.8 \times 10^{-7}$ N (a very small force).

Gravity on Other Worlds

Weight and Acceleration Due to Gravity

Weight depends on the local gravitational acceleration, which varies with the mass and radius of the planet or moon.

• Weight on a Planet: $W = m g$

• Surface Gravity: $g = G \frac{M}{R^2}$, where $M$ is the planet's mass and $R$ its radius.

Example: An 80-kg astronaut on the Moon (where $g_{\text{moon}} = 1.62$ m/s2) weighs $W = 80 \times 1.62 = 129.6$ N, much less than on Earth.

Gravity and Orbits

Satellites in Circular Orbits

Satellites in circular orbits are kept in motion by the gravitational force, which provides the necessary centripetal force.

• Orbital Speed: $v = \sqrt{\frac{GM}{r}}$

• Orbital Period: $T = \frac{2\pi r}{v}$

• Combining the above gives: $T^2 = \frac{4\pi^2}{GM} r^3$

Example: The radius of a geostationary satellite's orbit (period $T = 24$ hours) can be found by rearranging the above equation.

Rotational Motion

Describing Rotational Motion

Rotational motion involves objects spinning about an axis. Key variables include angular position, angular displacement, angular velocity, and angular acceleration.

• Angular Position ($\theta$): The angle (in radians) from a reference axis to the position of a point on the rotating object.

• Angular Displacement ($\Delta \theta$): The change in angular position: $\Delta \theta = \theta_2 - \theta_1$

• Angular Velocity ($\omega$): The rate of change of angular position: $\omega = \frac{\Delta \theta}{\Delta t}$

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

• 1 revolution = $2\pi$ radians = 360°

Relationship between Linear and Angular Quantities:

• Linear speed: $v = \omega r$

• Tangential acceleration: $a_t = \alpha r$

Torque and Rotational Dynamics

Torque

Torque is the rotational equivalent of force, determining how effectively a force causes an object to rotate about a pivot.

• Definition: $\tau = r F \sin \phi$, where $r$ is the distance from the pivot, $F$ is the force, and $\phi$ is the angle between $r$ and $F$.

• Units: Newton-meter (N·m)

• Moment Arm: The perpendicular distance from the pivot to the line of action of the force.

• Torque is positive for counterclockwise rotation, negative for clockwise.

Net Torque: The sum of all individual torques acting on an object.

Newton's Second Law for Rotation

• Net torque causes angular acceleration: $\tau_{\text{net}} = I \alpha$, where $I$ is the moment of inertia.

• This is analogous to $F = m a$ for linear motion.

Summary Table: Linear vs. Rotational Motion

Linear Motion	Rotational Motion
Displacement: $x$	Angular Displacement: $\theta$
Velocity: $v$	Angular Velocity: $\omega$
Acceleration: $a$	Angular Acceleration: $\alpha$
Force: $F$	Torque: $\tau$
Mass: $m$	Moment of Inertia: $I$
$F = m a$	$\tau = I \alpha$

Key Equations

• $f = \frac{1}{T}$

• $v = 2\pi r f = \frac{2\pi r}{T}$

• $a = \frac{v^2}{r}$

• $F = m \frac{v^2}{r}$

• $F = G \frac{m_1 m_2}{r^2}$

• $g = G \frac{M}{R^2}$

• $v = \sqrt{\frac{GM}{r}}$

• $T^2 = \frac{4\pi^2}{GM} r^3$

• $v = \omega r$

• $a_t = \alpha r$

• $\tau = r F \sin \phi$

• $\tau_{\text{net}} = I \alpha$

Additional info: These notes synthesize and expand upon the provided lecture slides and text, adding definitions, examples, and a summary table for clarity and completeness.