Physics: Uniform Circular Motion

Uniform Circular Motion

• Uniform circular motion is movement in a circular path at constant speed.
• Pure rotational motion is when all points in an object move in circular paths centered on a single point.
• Pure translational motion is motion with no rotation.

Rotation Angle and Angular Velocity

• Kinematics studies motion along a straight line, including displacement, velocity, and acceleration.
• Uniform circular motion is described using angular quantities.
• Rotation angle is the ratio of arc length to the radius of curvature.
• ΔΘ = AS/r, where ΔΘ is the rotation angle, AS is the arc length, and r is the radius. 
• One complete revolution has a rotation angle of 2π radians.
• 1 radian is equal to approximately 57.3 degrees
• Angular velocity (ω) is the rate of change of an angle: ΔΘ / Δt.

Angular Velocity

• Describes how fast an object rotates.
• ω = ΔΘ / Δt, where ω is angular velocity, ΔΘ is the change in angle, and Δt is the change in time.
• Radians per second (rad/s) are the units of angular velocity.
• Greater change in angle in a given time means greater angular velocity.

Relationship Between Angular and Linear Velocity

• Linear velocity (v) is the rate of change of arc length: AS/ Δt.
• v = rω, where v is linear velocity, r is the radius and ω is angular velocity.
• Linear velocity is proportional to the distance from the center of rotation, so the largest velocity will be on the rim (outer edge) of a rotating object.

Centripetal Acceleration

• Acceleration is a change in velocity, in magnitude or direction, or both.
• In uniform circular motion, the direction of velocity changes constantly, meaning there is always acceleration.
• This acceleration is called centripetal acceleration (ac).
• Centripetal acceleration points directly toward the center of rotation.
• Magnitude of centripetal acceleration: ac = v²/r = rω². (v is linear velocity, r is radius, ω is angular velocity)

Centripetal Force

• Any force that causes uniform circular motion is called a centripetal force.
• Centripetal force points toward the center of curvature, same direction as centripetal acceleration.
• Fc = mac = mv²/r = mrω². (Fc is centripetal force, m is mass, a is acceleration, v is linear velocity, r is radius, ω is angular velocity)
• Centripetal force is always perpendicular to the path and to the velocity, pointing to the center of curvature.

Banked Curves

• Road banking helps negotiate curves without friction. 
• Greater bank angles allow for faster speeds.
• "ideally banked curve," the angle is such that a vehicle can negotiate the curve at a certain speed without the assistance of friction.  
• For ideal banking, the net external force is equal to the horizontal centripetal force in the absence of friction.
• The components of the normal force in the horizontal and vertical directions must equal the centripetal force and the weight of the object, respectively
• N sin θ = mv²/r and N cos θ = mg where N is the normal force, θ is banking angle, m is mass, v is speed, r is radius, and g is acceleration due to gravity.

Newton's Universal Law of Gravitation

• Gravity is a fundamental force, acting at a distance.
• It's the weakest force in nature.
• All objects in the universe attract each other. 
• The force is directly proportional to the product of the masses and inversely proportional to the square of the distance: F = G(mM/r^2), where G is the universal gravitational constant, m and M are the masses, and r is the distance.
• G ≈ 6.674 x 10⁻¹¹ N⋅m²/kg².

Tides

• Ocean tides are a result of the Moon's gravitational pull.
• A high tide is on the side of Earth facing the Moon because the gravitational pull is strongest there.
• There is also a high tide on the opposite side of Earth. This occurs because Earth is pulled more strongly toward the Moon than the water is, which is further away.
• Earth's rotation leads to two high tides and two low tides per day.
• The sun also affects tides, though less significantly than the moon. Spring tides occur when the Earth, Moon, and Sun are aligned, leading to larger tidal ranges. Neap tides occur when the Sun is at a 90-degree angle to the Earth-Moon alignment, leading to smaller tidal ranges.

Weightlessness and Microgravity

• Weightlessness is experienced when the apparent net acceleration of an object is very small compared to that produced by Earth at its surface.
• It's not the absence of gravitational force. It's a state where the gravitational force produces the acceleration in freefall.
• When standing, most of your blood is below your heart, but in a horizontal position the opposite is true.
• Microgravity is found in space.

Cavendish Experiment

• Experiment conducted to determine the universal gravitational constant(G).
• Measured tiny gravitational attraction between ordinary-sized masses. 
• Determined the value of G to high accuracy.