Circular Motion

A car turning on a level road

• A car might be thrown outwards due to insufficient centripetal force.

Work done by the centripetal force

• The work done by the centripetal force on a body of mass m moving with velocity v in uniform circular motion through half the circular path is zero.

Uniform circular motion

• In uniform circular motion, both acceleration and velocity change.
• It's accelerated motion due to the change in velocity.

Passengers in a car

• Passengers in a car are thrown outwards when the car negotiates a curve.
• Cyclists bend inwards to counteract the centrifugal force.

Torque and centripetal force

• Torque produced by the centripetal force F in a circle of radius r is zero.

Angle

• The angle between radius vector and centripetal acceleration is π rad.

Angular displacement

• Angular displacement in circular motion is a dimensionless quantity.

Circular Motion Acceleration

• Acceleration cannot be predicted from the provided information, for a particle moving along a circular path of constant radius.

Kinetic energy of a body

• The kinetic energy of a body of mass m moving in a horizontal circle of radius 'r' with centripetal force F is F.r/2.

Momentum change

• When a particle of mass m moves in a horizontal circle of radius r with uniform speed v, its momentum changes by 2mv when it moves from one point to a diametrically opposite point.

Direction of angular displacement

• The direction of angular displacement in Uniform Circular Motion (U.C.M.) is given by either the left-hand rule or the right-handed screw rule.

Radial acceleration

• A particle moving on a circular path with constant speed possesses radial acceleration.

Force and circular motion

• A particle acted upon by a force of constant magnitude, which is always perpendicular to the velocity, will undergo circular motion.

Angular acceleration

• In uniform circular motion, angular acceleration is the rate of change of angular velocity with respect to time.

Acceleration relation

• During non-uniform circular motion, there is no specific relation between tangential acceleration (aT) and centripetal acceleration (ac).

Kinetic energy of a body

• For a body moving with uniform speed in a horizontal circle of radius r under a centripetal force k/r, the kinetic energy (K.E.) of the body is k / 2r, where k is a constant.

Satellite behavior

• If the gravitational force suddenly disappears for a satellite revolving around Earth with uniform speed v, the satellite moves with velocity v tangentially to its orbit.

Angular displacement vector

• Infinitesimal angular displacement of a particle performing uniform circular motion is a vector because it obeys the cumulative and associative laws of vector addition.

Uniform rotatory motion

• In uniform rotatory motion along a circular path, the magnitude of acceleration is constant.

Angular speed example

• For a body of mass 10 g tied to a string 20 cm long and whirled in a horizontal circle, the maximum angular speed is N.

Constant speed acceleration

• A particle revolving around a circular path with constant speed possesses acceleration along the radius.

Angular speed and time period

• If a particle goes around a circle once in a time period T, the angular velocity is 2π/T.

Path nature and velocity

• The nature of the path followed by a particle is determined by its velocity.

Angle of banking

• For a body moving in a circular path of radius r with uniform speed, the angle of banking is given by tan⁻¹(v² / r.g).

Wheel Rotating

• When a wheel rotates about an axis passing through the center and perpendicular to the plane with slowly increasing angular speed, it has tangential velocity and radial components.

Linear angular velocity relation

• The relation between linear velocity and angular velocity of a body moving in a circle in vector form is v = ω × r.

Centripetal and Centrifugal Force

• Centripetal force is directed opposite to the centrifugal force.

String tension

• When a 1kg body tied to the end of a string 1m long is whirled in a vertical circle with a constant speed of 4 m/s, the tension in the string is 6.2 N at the top of the circle.

Work Completed

• The work done by the centripetal force F when a body completes one revolution around a circle of radius R is zero.

Angular displacement of minute hand

• The angular displacement of the minute hand in 20 minutes is 2π/3 rad.

Scooter

• A scooter going around a circular road of radius 100m at a speed of 10 m/s has an angular speed of 0.1 rad/s.

String Mass Speed

• For a body of mass m tied to a string of length r at its lowest position, the minimum speed to complete one revolution is √5gr.

Constant in UCM

• All of the following remain constant in UCM: speed, kinetic energy, and angular momentum.

Car Traveling Acceleration

• A motor car traveling at 20 m/s on a circular road of radius 400 m, increases its speed at 1m/s²; its acceleration is √2 m/s².

Circular Path

• To enable a particle to describe a circular path, the angle between its velocity and acceleration should be 90°.

Friction and Height

• For a body sliding without friction along an inclined plane of height 'h' to loop a loop of radius 'r' at the bottom, the value of height is h = 5r / 2.

Earth and Circular Motion

• Earth is slightly bulged at the equator (oblate spheroid) because centrifugal force is maximum at the equator.

String Tension

• If a stone tied to a string is whirled in a circle, the tension in the string provides with the centripetal force.

Magnitudes and Vectors

• All Magnitudes of the following can be treated as Vectors: angular displacement and average angular velocity.

Simple Pendulum

• For a simple pendulum of length l, the minimum velocity to impart to its bob at the mean position, to reach a height equal to l above the point of suspension, is √5gl.

Radius Vector

• The scalar product of the radius vector and the tangential velocity is zero.

Centrifugal Force

• Centrifugal force is a pseudo force; it arises due to the accelerated frame of reference.

Flywheel

• A flywheel rotating at a constant speed of 3000 rpm describes an angle of 100π radians in one second.

String Breaks

• If a stone tied to a string is whirled, the string breaks at a certain speed because the required centripetal force is greater than the tension sustainable by the string.

Driver and Broad Wall

• A driver of a car traveling at velocity “v” suddenly sees a broad wall ahead and must brake sharply to avoid it.

Angular Speed

• The angular speed of the hour hand of the clock is none of the options provided in the question (2π rad/day).

Motor Cyclist

• For a motor-cyclist in a circus riding along a circular track of radius r in a vertical circle, the minimum speed at the highest point to avoid a fall is √gr.

Centripetal Force

• The centripetal force is a real force which provides the real interacting force of all the options that are given like mechanical force etc.

Minute Hand

• The angular speed of the minute hand of the clock in degrees per second is 0.1.

Circular Road

• Given a cyclist moving around a circular road of radius 50 meters, with µ = 0.2, the maximum velocity without skidding = 10 m/s.

Stone Mass

• Given a stone of mass “m,” tied to a string of length “L,” and rotated in a circle at constant speed “v,” if the string is released, the stone flies tangentially.

Scooter

• Given a scooter going around a track with 20 m/s at a 50-meter radius, angular velocity = 0.4 rad/s.

Cyclist Doubling

• A cyclist doubling speed results in increases 4x the force by which he leans with vertical.

Turn Table

• A spirit level placed at the edge of a turn table along its radius will point at Any point.

Radian

• One radian equals 57°.

Particle

• A particle covers equal distances in equal intervals of time; uniform non-zero rate of angular displacement.

Rotation

• A cyclist leans inwards when they take a circular turn.
• A ball of mass 0.25kg attached to the end of a string of length 1.96 m is moving in a horizontal circle.
• If the tension is more than 25 N. 14 m/s is maximum speed with which the ball can be moved.
• The highest point is where the particle has tension equal to zero.
• A particle moving around a circular path with uniform angular speed w has an acceleration is the radius.
• For a stone of mass m rotating in a circle, the force of tension trying to break the string is 𝑚𝑟𝜔².
• The string of a conical pendulum will become horizontal when: speed of revolution is almost infinite
• For a particle moving at a circular path of radius r at a uniform speed v, the angle described by the particle in one second is given by v/r.
• The string of a conical pendulum will become horizontal when the speed of revolution is almost infinite.
• If a wheel rotates about an axis passing through the center and perpendicular to the plane with slowly increasing angular speed, it has both tangential velocity and radial components.
• The relation between linear velocity and angular velocity of a body moving in a circle in vector form is v = ω × r.
• Centripetal force is directed opposite to the centrifugal force.
• A 1 kg body tied to a string 1 m long is whirled in a vertical circle with a constant speed of 4 m/s, the tension in the string is 6.2 N at the top of the circle.

Machines and Circular Motion

• Centrifugal machines use centrifugal force for their action.
• The normal force on a motorcycle ascending an overbridge decreases
• The angular acceleration of a particle moving along a circular path with uniform speed is zero.
• Cream is separated from milk because cream is lighter than skimmed milk.
• The nature of the path followed by the particle is determined by its velocity.
• A body of mass 10 kg moving in a circle of radius 1 m with an angular velocity of 2 rad/s experiences a centripetal force of 40 N.
• A particle describing a circular path of radius 10m every 2s has an average angular speed during 4s is None of the above.
• A man going on a cycle round a circular track of length 34.3 meters completes one round in √22 sec. The angle through which he leans inward is 45°.
• If a particle moves in a circle describing equal angles in equal times in a plane, its velocity changes in direction.
• A particle moving along a circular path with uniform speed has an instantaneous velocity and acceleration at 90°.
• The string of a conical pendulum will become horizontal when the speed of revolution is almost infinite.
• If 14 m s⁻¹ is the velocity an iron ball passes from the lowest position when it is moving from rim to rim. the bottom of a small hemispherical bowl radius of 10cm.
• A motorcyclist rides a round vertical wall falls down because The frictional force balances his weight.
• If a car moves on a circular road, it describes equal angles about the centre in equal intervals of time.
• For a car moving on a curved bridge of radius “r," The maximum velocity with which it can move round it without leaving the bridge at its highest point is v = √(gr)
• At banked curve road greater than weight for A car moving along the horizontal banked curved road.
• A particle revolving in a circle, clockwise in the plane of the paper has angular acceleration directed Perpendicular to the plane of the paper.
• A particle moving along a circular path with uniform speed, the angle of change of angular velocity = 0°.
• One kg stone tied to a meter long string is whirled in a vertical circle with 4 m s⁻¹ has a 6 Newton tension in the spring.

Circular Motion

• A body moving in a circular path with constant speed has variable acceleration.
• A bucket full of water rotated in a vertical circle doesn’t spill because the water provides the necessary centripetal force.
• If a body of mass 1000 gm is tied to the end of length 100 cm, whirled in a horizontal circle second, then tensionin the string= 4π²N.
• If fingers of the right hand are curled in the sense of motion of a particle performing circular motion then the outstretched thumb gives the direction of angular velocity and acceleration.

Speed

• Is length increases In angular speed of the watch after 15 seconds when The length of the second hand of a watch is 10mm. Is zero
• For a car moving along a circular road of radius r and coefficient of friction μ then maximum velocity is v = √μrg
• An important consequence of centrifugal force is that the earth is oblate spheroid.

Velocity

• A .25 m (g = 9.8 m/ s²) a coin that a coin kept on a horizontal, rotating has centre with 0.1m/s the angular velocity of the disc at which the coin will slip off, ( g= 9.8 m/$2) is
• Two satellites are going around the earth, if angular speed of both is same, than the centripetal acceleration will be more for second
• If 4.1 cm Is the amount lift required to A scooterist to bend on 480M. radius with 72/hr
• The angular velocity for a .5 wheel when 35 m is liner
• .443 m/s when A pendulum bob, 60 degrees is displace for released
• Constant is a type of Speed or when A particle covers equal distances around a path

Bodies in Motion

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• If a cyclist Is in motion Is 2x
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Circular Motion and Tension

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• The string=110, When its position :6, tension: 24.1N
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• .6-2, With a when radius + 2, than equal, circle force tension: 5 * 10-2.
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Dynamics of Circular Motion

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• , If when *2 equal when /4.

Rotational Motion Basics

• Rotational motion: objects spin/rotate around an axis.
• Linear motion: objects move forward in a straight line (aka translational motion).
• Linear and translational motion are essentially the same.

Angular Position and Displacement

• Angular position: point on a circle defining the object's rotational location.
• Angular displacement: change in angular position as an object rotates (delta theta).
• Angular displacement is calculated as final angular position - initial angular position.
• Radians are the standard unit for angular displacement.
• Angular position is a point on a circle, akin to position in linear motion.
• Angular displacement is the change in angular position, represented as Δθ.
• Δθ = final angular position - initial angular position.
• Standard unit for angular displacement is radians.

Angular Velocity

• Angular velocity (ω): measures how fast an object is spinning on a circle.
• Average angular velocity: angular displacement divided by time.
• Units for angular velocity: radians per second.
• Linear velocity (v) is related to angular velocity: v = ω * r, where r is the radius.
• Angular velocity (ω) measures how fast an object spins on a circle.
• Linear velocity measures how fast an object moves forward.
• Average linear speed = distance / time (meters/second).
• Average angular velocity (ω) = angular displacement / time (radians/second).
• Linear velocity (v) is related to angular velocity (ω) by the equation: v = ω * r, where r is the radius.

Angular vs Linear Speed

• Angular speed: the same at every point on a spinning object.
• Linear speed: not the same; points farther from the center travel a longer distance in the same time, and thus have a greater linear speed.
• Angular speed: the same at every point on a spinning object.
• Linear speed: not the same; points farther from the center travel a longer distance in the same time, and thus have a greater linear speed.
• If a wheel spins at 5 radians per second, all points have the same angular velocity.
• A point further from the center (point B) has a greater linear speed than a point closer to the center (point A).
• This is because point B has to travel a longer distance to maintain the same angular speed.

Period and Frequency

• Period (T): time to complete one full cycle: T = total time / number of cycles.
• Frequency (f): number of cycles per second: f = number of cycles / total time.
• Period unit: seconds per cycle.
• Frequency unit: Hertz (Hz) or 1/seconds.
• Period is the reciprocal of frequency: T = 1/f.
• Angular velocity can be calculated from frequency: ω = 2πf.
• Angular velocity can be calculated from the period: ω = 2π / T.
• Period (T) is the time to complete one cycle or rotation.
• T = total time / number of cycles (seconds per cycle).
• Frequency (f) is the number of cycles per second, the reciprocal of the period.
• f = number of cycles / time (Hertz or 1/seconds).
• Relationship: f = 1 / T.
• Angular velocity can be calculated using frequency: ω = 2πf.
• Angular velocity can be calculated using period: ω = 2π / T.

Angular Acceleration

• Linear acceleration: the change in velocity divided by the change in time.
• Angular acceleration is the change in angular velocity divided by the change in time.
• Units for linear acceleration: meters per second squared (m/s²).
• Units for angular acceleration: radians per second squared (rad/s²).
• Linear acceleration is the change in velocity over time.
• Average angular acceleration (α) = change in angular velocity / change in time.
• Units for linear acceleration: meters/second².
• Units for angular acceleration: radians/second².

Centripetal Acceleration

• Centripetal acceleration (ac): any object moving in circular motion has this, and it points towards the center of the circle.
• Centripetal acceleration calculation: ac = v²/r, or ac = ω² * r.
• If an object moves at a constant speed around a circle, its net acceleration equals the centripetal acceleration.
• Centripetal acceleration (ac) arises when an object moves in circular motion.
• It points towards the center of the circle.
• ac = v² / r, where v is linear speed and r is the radius.
• ac can also be expressed as ac = ω² * r, where ω is angular velocity.
• If an object moves with constant speed in a circle, the net acceleration equals the centripetal acceleration.

Tangential Acceleration

• Tangential acceleration (at): object is accelerating around a circle (not constant speed).
• Tangential acceleration calculation: at = α * r, where α is the angular acceleration.
• Tangential acceleration is also calculated as the change in velocity divided by the change in time.
• Tangential acceleration occurs if an object's speed changes as it moves around a circle.
• Tangential acceleration (at) = angular acceleration (α) * r.
• Tangential acceleration can also be found using at = change in velocity / change in time.

Net Acceleration

• Net acceleration is the vector sum of centripetal and tangential acceleration when the object is not moving at a constant speed.
• When an object accelerates around a circle, it has both centripetal and tangential acceleration.
• These accelerations are perpendicular to each other.
• The net acceleration is the vector sum of the centripetal and tangential accelerations.
• It can be visualized as the hypotenuse of a right triangle formed by the two acceleration vectors.