Circular Motion, Centripetal Force, and Dynamics, and Non-Uniform Acceleration Study Guide

Instantaneous Velocity in Circular Motion: For an object moving in a circle, the instantaneous velocity is always tangent to the circle at any given point. * Tangential Velocity: This instantaneous velocity is commonly referred to in circular motion as the tangential velocity. * Directional Examples (Clockwise motion): * Top of the circle: The velocity vector points straight to the right. * Right side of the circle: The velocity vector points straight down. * Bottom of the circle: The velocity vector points straight to the left. * Linear Path Realization: If the constraint holding the object (such as a string) were suddenly cut or snapped at a specific instant, the object would immediately travel in a straight line in the direction of the tangential velocity at that moment.

Definition of Centripetal Acceleration ( $a_c$ ): This is the acceleration that keeps an object on its circular path. It is directed toward the center of the circle at every point in the motion. * Example: At the top of a clockwise circle, the velocity is toward the right, but the centripetal acceleration points downward (toward the center), forcing the object to curve.
Definition of Centripetal Force ( $F_c$ ): Consistent with Newton's Second Law ( $F = ma$ ), there must be a force accompanying the centripetal acceleration. This is defined as: * $F_c = ma_c$
Role of Centripetal Force: It is important to note that centripetal force is not a "specific" force (like gravity or tension) but a category. A specific force must be responsible for providing the centripetal motion. * Examples of providers: Tension in a string, gravity for a satellite, or the normal force on a track.

Characteristics of UCM: This occurs when an object travels around a circular path at a constant speed.
Velocity vs. Speed in UCM: * Even if the speed (magnitude) is constant, the velocity is not constant because the direction of the vector is continuously changing. * Because velocity is changing, there is a non-zero acceleration.
Mathematical Formulas for UCM: * Centripetal Acceleration: * $a_c = \frac{v^2}{r}$ * Centripetal Force: * $F_c = \frac{mv^2}{r}$ * Variables: $v$ is tangential velocity, $r$ is the radius of the circle, and $m$ is the mass of the object.

Sum of Forces: When solving problems involving the direction pointing toward or away from the center of a circle, the sum of the forces ( $\sum F$ ) totals to the centripetal force ( $ma_c$ or $\frac{mv^2}{r}$ ).
Sign Conventions (Instructional Preference): * Technically, standard physics convention often defines the direction away from the center as positive and toward the center as negative (making the acceleration $- \frac{v^2}{r}$ ). * Simplified Convention: To reduce student error, the instructor defines forces pointing toward the center of the circle as positive and forces pointing away from the center as negative. This matches the positive nature of the centripetal force ( $\frac{mv^2}{r}$ ) in the calculation.

Tangential Acceleration ( $a_t$ ): This occurs if the linear speed of the object is changing (speeding up or slowing down) as it moves around the circle. * Formula: $a_t = r \times \alpha$ * Variables: $r$ is the radius and $\alpha$ is the angular acceleration (measured in $rad/s^2$ ). * Direction: The tangential acceleration points in the same direction as the tangential velocity.
Total Acceleration ( $a_{total}$ ): In non-uniform motion, an object experiences both centripetal ( $a_c$ ) and tangential acceleration ( $a_t$ ). * These two components are always perpendicular ( $90^\circ$ apart). * Total Acceleration Formula (Pythagorean Theorem): * $a_{total} = \sqrt{a_t^2 + a_c^2}$

Given Data: * Environment: Space (gravity $g = 0$ ). * Mass ( $m$ ): $0.25\,kg$ . * Radius ( $r$ ): $2.0\,m$ (length of string). * Tangential Speed ( $v$ ): $4.0\,m/s$ (constant).
Calculations: 1. Centripetal Acceleration ( $a_c$ ): * $a_c = \frac{v^2}{r} = \frac{4.0^2}{2.0} = \frac{16}{2} = 8.0\,m/s^2$ 2. Centripetal Force ( $F_c$ ): * $F_c = m \times a_c = 0.25 \times 8.0 = 2.0\,N$ 3. Tension in the String ( $T$ ): * Setup: $\sum F = ma_c$ * The only force acting toward the center is tension ( $T$ ). Because gravity is zero, weight is not considered. * $T = F_c = 2.0\,N$

Given Data: * Mass ( $m$ ): $500\,kg$ . * Radius ( $r$ ): $10\,m$ . * Constant Speed ( $v$ ): $20\,m/s$ .
Concepts of Apparent Weight: * The normal force ( $N$ ) is what a person feels as their weight. A higher normal force makes a rider feel heavier; a lower normal force makes them feel lighter. * A rider feels heaviest at the bottom of the loop and lightest at the top.
Calculations for Specific Points: 1. Bottom of the Loop: * Free Body Diagram: Normal force ( $N$ ) points up (toward center), Weight ( $mg$ ) points down (away from center). * Equation: $N - mg = \frac{mv^2}{r}$ * $N = \frac{mv^2}{r} + mg$ * $N = \frac{500 \times 20^2}{10} + (500 \times 9.8) = 20,000 + 4,900 = 24,900\,N$ . 2. Top of the Loop: * Free Body Diagram: Both Normal force ( $N$ ) and Weight ( $mg$ ) point down (toward center). * Equation: $N + mg = \frac{mv^2}{r}$ * $N = \frac{mv^2}{r} - mg$ * $N = 20,000 - 4,900 = 15,100\,N$ . 3. Side of the Loop: * Free Body Diagram: Normal force ( $N$ ) points toward center. Weight ( $mg$ ) points down (perpendicular to the centripetal direction). * Equation for centripetal direction: $N = \frac{mv^2}{r}$ * $N = 20,000\,N$ (Expressed as $2.00 \times 10^4\,N$ for significant figures).
Theoretical Note: Maintaining constant speed in such a loop requires a "modulating force" to counteract changes in the weight's component relative to the direction of motion; otherwise, the car would naturally slow down while ascending.

Given Data: * Radius ( $r$ ): $0.25\,m$ . * Initial State: Rest ( $v_0 = 0$ ). * Angular Acceleration ( $\alpha$ ): $12\,rad/s^2$ . * Time ( $t$ ): $0.33\,s$ .
Calculations for Tangential Acceleration: * $a_t = r \times \alpha = 0.25 \times 12 = 3.0\,m/s^2$
Context for Total Acceleration: * To find the total acceleration at $t = 0.33\,s$ , the centripetal acceleration ( $a_c = \frac{v^2}{r}$ ) must also be calculated for that specific instant. This requires finding the instantaneous tangential velocity ( $v$ ) using angular kinematics before applying the Pythagorean theorem.