If Any object moves such that it covers an angle q at the fixed point (center of the circle) and its gap from the fixed point remains constant (radius of the circle)

Here R is constants

Now finding linear speed of

(i) hour hand (30 cm)

(ii) Minute hand (90 cm)

(iii) Second hand (60 cm)

Solution:- (i) time period of hour hand T = 12 hr

= 12 x 3600 sec

v = r w

(ii) Minute hand:- time period = T = 60 min = 60 x 60 sec

(iii) Second hand:-

Time period= T = 60 sec

v = r w

**Types of circular motion: (on the basis of speed)**

(1) Uniform circular motion:** (U C M)**

Only direction of velocity is changing, magnitude remains unchanged.

Now acceleration due to change in direction of velocity

Using law of parallelogram

= 2v^{2} - 2v^{2} cosq

= 2v^{2 }(1- cosq)

Now

sinq » q if q <<<<

a = v w,

Direction of this acceleration is towards center

Centripetal acceleration or Radial acceleration

(2) Non uniform circular motion (Non U C M)

Here in this case direction as well as magnitude both are changing continuously therefore here two different named acceleration will be as,

(i) a_{c} = a_{R} (centripetal or radial acceleration)

(ii) a_{t} (Tangential acceleration)

Now,

**Tangential acceleration. **

**“**Rate of change of magnitude of velocity”

v = R w

So a_{t} ^ a_{c}

Net acceleration for non-uniform circular motion.

**Example:-** A particle moving in a circular path of radius 2 meter and its velocity varies as v = 10t^{2.} Then net acceleration of the particle at t = 2 sec.

**Solution:** Given v = 10t^{2}

= 20 x 2

= 40 m/s^{2}

**Centripetal force:-** the force require to move in a circular path for any object with respect to inertial frame is called centripetal force.

**Note:-** Centrifugal force will also have same magnitude but direction opposite to centripetal force.

Direction of this force is towards the center.

**HORIZONTAL CIRCULAR MOTION**

**(1) Only banking**

Here N sinq is C.P. provider

But N cosq = mg

Safe speed

**(2) Only friction **

Here now m N is C.P. provider

N = mg

Safe speed

**Friction and Banking both:- **To prevent inward sliding

cosq + m N sinq = mg

Divide

**TO PREVENT OUTWARD SLIDING**

N cosq - m N sinq = mg

v_{1} < V_{safe} < v_{2}

**CONICAL PENDULUM**

T sinq = mrw^{2}

T cosq = mg

Time period of conical pendulum

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