When a body moves along a circular path with uniform speed, it is said to perform a uniform circular motion (U.C.M.).
1. Angular speed of a particle,
Where θ is the angular displacement in time‘t’. T is the period of revolution of the particle performing a U.C.M. and n is the frequency of revolution i.e. no .of revolutions completed by the particle per second. S.I. unit of ω is rad/s.
Angular Velocity (ω): It is the rate of change of angular displacement w.r.t. time.
2. The linear speed of a particle v = rω where r is the radius of the circle.
3. S.I. unit of α is rad/sec2
Linear acceleration (a) = r α.
4. In U.C.M. the acceleration is always directed along the radius, towards the centre of the circle. It is called radial or centripetal acceleration. The magnitude of C.P. acceleration is . If m is the mass of the body performing U.C.M., then the centripetal force acting upon it is . It is also given by
The radial acceleration
5. If the angular acceleration α is constant then for accelerated circular motion,
Where ω0 is the angular speed of the particle at time t = 0 and ω is the angular speed of the particle after time‘t’.
6. Banking of Roads: For a vehicle moving along a circular track of radius r, banked at an angle θ, the maximum speed limit for the safety of the vehicle is given by
V is also known as critical speed. In the case of a train, the elevation of the outer rail above the inner rail is given by
h = ℓ sin θ.
Where is the distance between the rails.
For a metre gauge railway,
7. For a vehicle moving along a horizontal curve of radius r, (unbanked circular road), the C.P. force is provided by the friction. The maximum safe speed (v) is calculated by using the relation
Where μ is the coefficient of friction between the tyres of the car and the surface of the road.
8. A cyclist, moving with a speed v, along a horizontal curved road of radius r, has to lean inwards to keep his balance. His angle of inclination (θ) with the vertical is given by
11. Direction of are given by right hand thumb rule.
are axial vectors
12. When a particle performs a non uniform circular motion it has a centripetal acceleration and a tangential acceleration. directed along the tangent
13. As and are at right angles, the net acceleration
14. For a U.C.M. aT = 0 or α = 0. No work is done in U.C.M.
15. Motion in a Vertical circle:
When a body moves in a vertical circle,
· The tension (if string is used) or normal reaction at any point P is given by
· Tension at the lowest point (A) is maximum.
· Tension at the highest point (B) is minimum.
· The minimum horizontal velocity that should be given to the body at the bottom (A) to complete the vertical loop is given by
· Minimum speed required at
· Velocity at C or D =
· Tension at C or D (ends of the horizontal diameter)
· K.E. at A – K.E. at B = 2 mgr
· TA – TB = 6 mg
16. For looping a loop of radius r, the minimum height from which the body should be released is given by
17. The expressions, are used in the following cases:
(i) Bucket full of water rotated in a vertical circle.
(ii) Motor cyclist riding in a vertical circle in a hollow sphere.
18. Conical Pendulum:
(i) If T is the tension in the string, then
Where T is the period of the conical pendulum.
(iii) The period of a conical pendulum is less than the period of a simple pendulum of the same length.
19. Useful Points:
(i) Newton’s laws of motion are valid in an inertial frame of reference. In this frame, we consider the real forces.
(ii) A frame of reference which is moving with an acceleration w.r.t. the system of fixed stars is called a non-inertial frame or accelerated frame of reference. In this frame, Newton’s laws of motion are not valid. To apply Newton’s laws of motion to situations in non inertial frames, we consider the fictitious or pseudo forces. Centrifugal force is a pseudo force.
(iii) Centripetal force and centrifugal force do not form an action-reaction pair.
(iv) For a speed breaker,
(v) In banked roads, the horizontal components R sin θ provides the necessary centripetal force. While the vertical component R cos θ balances the weight (mg) of the car.