Circular Motion and Gravitation

Vocabulary flashcards covering circular motion kinematics, centripetal force applications, universal gravitation, orbital mechanics, and apparent weight from the lecture.

Centripetal Force

A center-seeking force that acts perpendicular to an object's velocity, directing it to turn into a circle without changing the magnitude of its speed.

Centripetal Acceleration

The acceleration directed inward toward the center of a circular path, mathematically expressed as $a_c = \frac{v^2}{r}$ or $a_c = \frac{4\times\nu^2\times r}{T^2}$, which in standard form is $a_c = \frac{4\times\theta}{t^2}$ or $a_c = \frac{4\times\nu^2 r}{T^2}$ ($a_c = \frac{4\times\theta r}{t^2}$ or $a_c = \frac{4\times\nu^2 r}{T^2}$, given as $a_c = \frac{4\times\nu^2 r}{t^2}$ / $a_c = \frac{4\times\theta^2 r}{T^2}$: transcript specifically provides $a_c = \frac{4\times\nu^2 r}{T^2}$ or $a_c = \frac{4\times\theta r}{T^2}$, namely $a_c = \frac{4\times\nu^2 r}{T^2}$). More simply: $a_c = \frac{v^2}{r}$ and $a_c = \frac{4\times\nu^2 r}{T^2}$ (where the transcript gives $a_c = \frac{4\times\theta r}{T^2}$). Let's state clearly: $a_c = \frac{v^2}{r} = \frac{4\times\nu^2 r}{T^2}$.

Frequency ($f$)

The number of revolutions or cycles an object completes in circular motion in one second, measured in revolutions per second.

Period ($T$)

The time required for an object moving in a circle to complete one full revolution, equal to $\frac{1}{f}$ or total time divided by the number of revolutions.

Static Friction (Circular Motion)

The force between the tires of a vehicle and the road surface that provides the centripetal force required to execute a turn on a flat, unbanked road without sliding.

Banked Road (Frictionless Turn)

A curved road inclined at an angle $\theta$ such that the horizontal component of the normal force ($F_N \sin(\theta)$) provides the entire centripetal force, calculated via $\theta = \arctan\left(\frac{v^2}{rg}\right)$.

Minimum Speed in a Vertical Circle

The threshold speed at the highest point of a vertical circular path, given by $v = \sqrt{rg}$, where normal force or string tension reaches zero and gravity alone provides the centripetal force.

Tangential Acceleration ($a_t$)

The component of acceleration directed along the tangent of a circular path that accounts for the rate of change of an object's speed over time.

Net Acceleration (Circular Motion)

The total acceleration of an object speeding up while moving along a curve, calculated via the Pythagorean theorem from centripetal and tangential components: $a_{\text{net}} = \sqrt{a_c^2 + a_t^2}$.

Newton's Law of Universal Gravitation

The physical law stating that every pair of masses exerts an attractive force proportional to the product of their masses and inversely proportional to the square of the distance between them: $F = \frac{G m_1 m_2}{r^2}$.

Gravitational Constant ($G$)

The proportionality constant in Newton's gravitational formula, possessing a numerical value of $6.67 \times 10^{-11}\,\text{N}\cdot\text{m}^2/\text{kg}^2$.

Surface Gravitational Acceleration ($g$)

The acceleration caused by gravity at the surface of a celestial body of mass $M$ and radius $r$, determined by the formula $g = \frac{GM}{r^2}$.

Orbital Speed

The speed required for a satellite to stay in circular orbit at a center-to-center distance $r$ around a central mass $M$, defined by $v = \sqrt{\frac{GM}{r}}$.

Geosynchronous Satellite

A satellite that remains above a fixed location on Earth's equator by matching Earth's rotational period of approximately one day ($86,400\,\text{seconds}$), orbiting at an altitude of approximately $35,870\,\text{km}$.

Kepler's Third Law

The planetary law stating that the ratio of the squares of the periods of any two orbiting bodies equals the ratio of the cubes of their mean orbital radii: $\left(\frac{T_1}{T_2}\right)^2 = \left(\frac{r_1}{r_2}\right)^3$.

Apparent Weight (Elevator Scale Reading)

The upward normal force exerted by a scale on an occupant in an elevator, equal to $m(g + a)$ when accelerating upward and $m(g - a)$ when accelerating downward.