Circular Motion and Angular Measurements

There are $2oldsymbol{ ext{π}}$ radians in a full circle.

Angular velocity ($oldsymbol{ ext{ω}}$) is the rate of change of angle measured in radians per second (rad/s).

The equation linking linear velocity ($oldsymbol{v}$) and angular velocity ($oldsymbol{ω}$) is $oldsymbol{v = ωr}$.

<p>Centripetal acceleration is the acceleration of an object moving in a circle, directed towards the center, measured in meters per second squared (m/s²).</p> Signup and view all the answers

<p>Centripetal force keeps an object moving in a circle, directed toward the center.</p> Signup and view all the answers

<p>Newton's first law states that an object moves at constant velocity unless acted upon by a force.</p> Signup and view all the answers

<p>The orbital period ($oldsymbol{T}$) can be calculated using $oldsymbol{ω = rac{2oldsymbol{π}}{T}}$.</p> Signup and view all the answers

<p>The relationship can be expressed as $oldsymbol{s = θr}$, where $s$ is the arc length, $θ$ is the angle in radians, and $r$ is the radius.</p> Signup and view all the answers

<p>Radians are the standard unit for measuring angles, providing a direct relationship between the angle and the arc length based on the radius.</p> Signup and view all the answers

<p>Linear velocity ($oldsymbol{v}$) is the speed of a particle moving along a circular path, measured in meters per second (m/s).</p> Signup and view all the answers

<p>Periodic time is the time taken for one complete cycle of motion, such as the time taken for a pendulum to swing once back and forth.</p> Signup and view all the answers

<p>Centripetal force occurs when an object moves in a circular path at a constant speed, as it continuously changes direction toward the center of the circle.</p> Signup and view all the answers

<p>Two examples of centripetal force are the gravitational force acting on the Moon while it orbits Earth and the tension in a string for a ball being swung in a circle.</p> Signup and view all the answers

<p>If the speed increases while the centripetal force remains constant, the radius of rotation R will also increase.</p> Signup and view all the answers

<p>Gravity is the force that keeps planets in orbit around the sun.</p> Signup and view all the answers

<p>If the orbital radius of the Moon were halved, its orbital velocity would increase by a factor of about $ ext{√2}$ according to $v^2 ∝ 1/R$.</p> Signup and view all the answers

<p>The period of orbit of a geostationary satellite is 24 hours.</p> Signup and view all the answers

<p>Geostationary satellites are used for communications and weather forecasting.</p> Signup and view all the answers

<p>The gravitational force and the centripetal force are equalized to derive Kepler’s Third Law.</p> Signup and view all the answers

<p>The equation is $T^2 = \frac{4\pi^2R^3}{GM}$, indicating the relationship between period and radius.</p> Signup and view all the answers

<p>Radians provide a direct relationship between the angle and the arc length, simplifying calculations in circular motion.</p> Signup and view all the answers

<p>Angular velocity indicates the rate of rotation, and it directly relates to linear velocity as $v = ωr$; this relationship is crucial in designing wheels for vehicles.</p> Signup and view all the answers

<p>Centripetal acceleration is constant in magnitude and directed towards the center of the circle, while linear acceleration can vary in both magnitude and direction; this distinction is important for analyzing circular motion dynamics.</p> Signup and view all the answers

<p>Periodic time is the duration for one complete cycle and is inversely related to frequency, which indicates how many cycles occur per unit time.</p> Signup and view all the answers

<p>Newton's first law states that an object will maintain its state of motion unless acted upon by a net external force, highlighting that centripetal force must continuously act on objects in circular motion to change their direction.</p> Signup and view all the answers

<p>As the speed of the pendulum bob increases, the radius of rotation also increases, demonstrating that higher velocity requires greater centripetal force to maintain circular motion.</p> Signup and view all the answers

<p>The gravitational force between the Earth and the Moon provides the necessary centripetal force that keeps the Moon in its nearly circular orbit.</p> Signup and view all the answers

<p>Kepler's Third Law states that the square of a satellite's orbital period is proportional to the cube of its orbital radius, reflecting the dependence of period on radius.</p> Signup and view all the answers

<p>If the Moon's orbital radius is halved, its orbital velocity would need to increase to maintain the balance with the gravitational force acting on it.</p> Signup and view all the answers

<p>Geostationary satellites maintain a fixed position over one point on Earth's equator, primarily used for communications and weather forecasting.</p> Signup and view all the answers

<p>Centripetal acceleration increases as the square of the speed increases, given that the radius remains constant.</p> Signup and view all the answers

<p>Angular velocity is a vector quantity because it has both magnitude and direction, indicating the axis of rotation and the rate of change of angle.</p> Signup and view all the answers

<p>The relationship is expressed as ω = 2π/T, where ω is angular velocity and T is the period.</p> Signup and view all the answers

<p>Increasing the radius results in an increase in linear velocity, as linear velocity is given by the formula v = ωr.</p> Signup and view all the answers

<p>Centripetal force is essential for keeping an object in circular motion, as it acts toward the center of the circle, preventing the object from flying off tangent.</p> Signup and view all the answers

<p>Periodic time is the time taken for one complete revolution in circular motion and is inversely related to frequency, which is the number of revolutions per second.</p> Signup and view all the answers

<p>Gravitational force provides the necessary centripetal force that acts toward the center of the orbiting path, enabling bodies to maintain their orbits.</p> Signup and view all the answers

<p>Understanding this relationship is crucial for applications like vehicle navigation and engineering, where precise speed and direction control is necessary.</p> Signup and view all the answers

<p>Halving the radius increases the satellite's orbital velocity, as velocity is inversely proportional to the square root of the radius.</p> Signup and view all the answers

<p>Uniform circular motion implies that an object travels in a circular path at a constant speed, although its direction is continuously changing.</p> Signup and view all the answers

<p>Newton's second law shows that gravitational force equals centripetal force, represented by the equation $F_{gravitational} = F_{centripetal}$.</p> Signup and view all the answers

<p>The orbital velocity is inversely proportional to the orbital radius; if the radius decreases, the velocity increases, as shown by $v^2 ∝ 1/R$.</p> Signup and view all the answers

<p>According to Kepler's Third Law, the period squared increases as the cube of the orbital radius increases: $T^2 ∝ R^3$.</p> Signup and view all the answers

<p>A geostationary satellite orbits at approximately 36,000 km, matching Earth's rotation, thus staying fixed over a specific point.</p> Signup and view all the answers

<p>Increasing the speed of the pendulum bob results in a larger radius of rotation, due to the relationship between centripetal force and velocity.</p> Signup and view all the answers

<p>The two forces are gravitational force and centripetal force, represented by the equations $F_{gravitational} = GMm/R^2$ and $F_{centripetal} = mv^2/R$.</p> Signup and view all the answers

<p>The centripetal force acting on the Moon is provided by gravitational attraction from the Earth, keeping it in a nearly circular path.</p> Signup and view all the answers

<p>The gravitational force is reduced to a quarter of its original value, following the inverse square law $F ∝ 1/R^2$.</p> Signup and view all the answers

<p>Varying the tension in the string alters the centripetal force, influencing the speed and radius of the pendulum bob's rotation.</p> Signup and view all the answers

<p>Angular velocity is related to the period by the formula $rac{2 ext{π}}{T}$, indicating how fast an object rotates around a circular path.</p> Signup and view all the answers