Motion in a Circle | Jamb(UTME)

Jamb(utme) key points on motion in a circle; angular velocity; angular acceleration

Motion in a Circle

1. Circular motion occurs when an object moves along a circular path.
2. The motion can be uniform (constant speed) or non-uniform (changing speed).
3. The radius $(r)$ is the distance from the center of the circle to the object in motion.
4. The period $(T)$ is the time taken for one complete revolution.
5. The frequency $(f)$ is the number of revolutions per second, measured in hertz $(Hz)$.
6. Frequency and period are inversely related:
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   $f = \frac{1}{T}$
7. The circumference of the circular path is:
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   $C = 2\pi r$
8. Centripetal acceleration is directed toward the center of the circle and keeps the object moving in a circular path:
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   $a_c = \frac{v^2}{r}$
9. Centripetal force provides the necessary force for circular motion:
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   $F_c = \frac{mv^2}{r}$
10. The centripetal force is not a separate force but can be gravitational, tension, frictional, or any other type of force.
11. Tangential velocity $(v)$ is the speed of the object along the circular path:
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    $v = \frac{2\pi r}{T}$
12. Circular motion has two components of acceleration: centripetal and tangential.
13. Tangential acceleration arises when the speed of the object changes:
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    $a_t = \frac{\Delta v}{\Delta t}$
14. Total acceleration combines centripetal and tangential acceleration:
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    $a_{total} = \sqrt{a_c^2 + a_t^2}$
15. Work done by centripetal force is zero because the force is perpendicular to the displacement.
16. The direction of motion in uniform circular motion constantly changes, but the speed remains constant.
17. Banking of roads and tracks ensures safe circular motion by providing additional centripetal force.
18. Objects in circular motion experience outward inertia, which is often mistaken as a "centrifugal force."
19. The time to complete one rotation in uniform circular motion is constant.
20. Applications of circular motion include satellites, Ferris wheels, and vehicles on curved roads.
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Angular Velocity

21. Angular velocity $(\omega)$ measures how fast an object rotates or revolves around a fixed axis.
22. It is defined as the rate of change of angular displacement:
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    $\omega = \frac{\theta}{t}$
23. The SI unit of angular velocity is radians per second $rad/s$.
24. One complete revolution corresponds to an angular displacement of $2\pi$ radians.
25. Angular velocity relates to linear velocity:
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    $v = r \omega$
26. Angular velocity is constant in uniform circular motion.
27. If an object makes $n$ revolutions per second, angular velocity is:
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    $\omega = 2\pi n$
28. Angular velocity is a vector quantity; its direction is given by the right-hand rule.
29. For an object rotating with f $Hz$, angular velocity is:
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    $\omega = 2\pi f$
30. Larger radii result in higher linear velocities for the same angular velocity.
31. Angular velocity remains the same for all points on a rigid body in rotation.
32. A decrease in the radius of motion increases angular velocity if angular momentum is conserved (e.g., a spinning ice skater pulling in their arms).
33. Uniform angular velocity implies equal angular displacement over equal time intervals.
34. In planetary motion, angular velocity decreases as the distance from the sun increases.
35. For a fixed radius, doubling the angular velocity doubles the linear velocity.
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Angular Acceleration

36. Angular acceleration $\alpha$ is the rate of change of angular velocity:
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    $\alpha = \frac{\Delta \omega}{\Delta t}$
37. The SI unit of angular acceleration is radians per second squared ${rad/s}^2$.
38. Angular acceleration occurs when the angular velocity of an object changes.
39. Angular acceleration can be caused by a change in speed, direction, or both.
40. If a wheel starts from rest, angular acceleration determines how quickly it reaches a given angular velocity.
41. For constant angular acceleration, angular displacement $\theta$ is:
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    $\theta = \omega_0 t + \frac{1}{2} \alpha t^2$
42. Final angular velocity $\omega_f$ is given by:
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    $\omega_f = \omega_0 + \alpha t$
43. The rotational analog of Newton’s Second Law relates torque $\tau$ to angular acceleration:
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    $\tau = I \alpha$
44. Angular acceleration is a vector quantity.
45. Positive angular acceleration increases angular velocity, while negative angular acceleration (angular deceleration) reduces it.
46. An increase in angular acceleration results in a steeper slope on an angular velocity-time graph.
47. Angular acceleration is uniform in systems with constant torque.
48. Applications of angular acceleration include gear systems, turbines, and flywheels.
49. In rotational motion, angular acceleration corresponds to the tangential acceleration:
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    $a_t = r \alpha$
50. Angular acceleration plays a key role in understanding systems where rotational speeds change, such as braking in vehicles or starting electric fans.
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Jamb(utme) key points on centripetal force; centrifugal force; and their applications

Centripetal Force

1. Centripetal force is the force that keeps an object moving in a circular path, directed toward the center of the circle.
2. The word "centripetal" means "center-seeking."
3. Centripetal force is not a new force; it is provided by forces such as tension, gravity, or friction, depending on the situation.
4. Formula for centripetal force:
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   $F_c = \frac{mv^2}{r}$ where $F_c$ is the centripetal force, $m$ is the mass, $v$ is the velocity, and $r$ is the radius.
5. Centripetal force is a vector quantity and always points toward the center of the circle.
6. For an object in uniform circular motion, centripetal force only changes the direction of velocity, not its magnitude.
7. In planetary motion, centripetal force is provided by gravitational attraction between the planet and the sun.
8. In cars turning on a road, centripetal force is provided by friction between the tires and the road.
9. A ball on a string moving in a circular path experiences centripetal force provided by the tension in the string.
10. Without centripetal force, an object would move in a straight line due to inertia.
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Centrifugal Force

11. Centrifugal force is a pseudo-force experienced in a rotating frame of reference, appearing to act outward from the center of rotation.
12. The word "centrifugal" means "center-fleeing."
13. Centrifugal force arises due to inertia in a rotating frame of reference.
14. It is not a real force but a perceived effect in non-inertial (rotating) frames.
15. In a car turning a corner, passengers feel they are pushed outward, which is due to centrifugal force.
16. Centrifugal force has the same magnitude but opposite direction as centripetal force in the rotating frame:
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    $F_{\text{centrifugal}} = \frac{mv^2}{r}$
17. Centrifugal force is used to explain phenomena in rotating systems, like washing machines or centrifuges.
18. The centrifugal effect is due to an object's tendency to maintain its linear velocity (inertia).
19. In a rotating carnival ride, the walls push inward to provide centripetal force, but riders feel an outward force due to centrifugal force.
20. Centrifugal force can create the illusion of weight in space-station designs.
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Differences Between Centripetal and Centrifugal Forces

21. Centripetal force is a real force, while centrifugal force is a pseudo-force.
22. Centripetal force is directed toward the center of the circle, while centrifugal force appears directed outward.
23. Centripetal force is required for circular motion, while centrifugal force is experienced in rotating reference frames.
24. Centripetal force keeps an object in circular motion, while centrifugal force explains the object's outward "push" in the rotating frame.
25. Centrifugal force is absent in inertial (non-rotating) reference frames.
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Applications of Centripetal Force

26. Banked Roads: Roads are tilted to provide the required centripetal force for vehicles to take turns safely.
27. Roller Coasters: Centripetal force keeps the cars on their curved tracks, especially during loops.
28. Satellites in Orbit: Gravitational force acts as the centripetal force, keeping satellites in circular or elliptical orbits.
29. Tethered Aerodynamic Experiments: Centripetal force acts on objects like a plane on a curved flight path.
30. Bicycle Turns: Friction between the tires and the road provides centripetal force for turning.
31. Pendulum Motion: A pendulum swinging in a circular arc experiences centripetal force due to the tension in the string.
32. Swinging Buckets: A bucket of water swung in a vertical circle stays intact because tension provides the centripetal force.
33. Earth’s Rotation: Centripetal force due to gravity prevents objects from flying off the Earth's surface as it rotates.
34. Electron Motion: Electrons orbiting a nucleus in an atom are kept in circular paths by centripetal electrostatic force.
35. Fluid Dynamics: Liquids in a rotating container experience centripetal force to maintain circular motion.
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Applications of Centrifugal Force

36. Centrifuges: Used to separate substances based on density by spinning samples at high speeds.
37. Washing Machines: Centrifugal force expels water from clothes during the spin cycle.
38. Amusement Rides: Rotating rides rely on centrifugal force to keep riders pressed against the walls.
39. Artificial Gravity: Space-station designs propose using centrifugal force to simulate gravity by rotating the structure.
40. Dryers: The spinning drum of a dryer uses centrifugal force to push water out of clothes.
41. Oil Separators: Industrial centrifuges separate oil and impurities using centrifugal force.
42. Milk Cream Separation: Cream is separated from milk using centrifugal force in dairy processing.
43. Gravitational Simulations: Centrifugal force is used in labs to simulate high-gravity environments.
44. Vehicles on Curves: Passengers experience centrifugal force pushing them outward during sharp turns.
45. Agricultural Equipment: Seed and fertilizer spreaders use centrifugal force to distribute material over fields.
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Combined Applications

46. Banked Tracks: The combined effects of friction, centripetal force, and centrifugal force allow vehicles to navigate curved tracks safely.
47. Saturn’s Rings: The balance between gravitational pull (centripetal force) and centrifugal tendencies keeps the rings stable.
48. Spinning Tops: The stability of a spinning top relies on the interplay of centripetal and centrifugal forces.
49. Water Pumps: Rotating blades create centrifugal force to push water through the system.
50. Circular Saw Blades: The stability and cutting action of rotating blades depend on the forces acting during circular motion.
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