**Angular and Circular Motion: Understanding Relationships**1. **Basic Concept**: When a rigid body rotates about a fixed axis, every particle in the body moves in a circular path. The arclength \( \Delta s \) between angular positions \( \theta_1 \) and \( \theta_2 \) is given by the equation \( \Delta s = r \Delta \theta \).2. **Assessment of Equation**: - (a) In the limit where \( \Delta \theta \) is very small, \( \Delta s \) can be considered as a straight line. True or false? - (b) Any point along the circular path has a tangential velocity (\( v_t \)) perpendicular to the radius. By dividing both sides of \( \Delta s = r \Delta \theta \) by \( \Delta t \), we get \( v_t = r \omega \), showing the relationship between angular velocity and linear velocity.3. **Change in Angular Velocity**: - (c) If angular velocity changes by \( \Delta \omega \), linear speed changes by \( \Delta v_t \). Hence, \( \Delta v_t = r \Delta \omega \). Is this true?4. **Angular and Tangential Acceleration**: - (d) If changes occur in some small \( \Delta t \), divide \( \Delta v_t = r \Delta \omega \) by \( \Delta t \) to get \( a_t = r \alpha \). Here, \( a_t \) is tangential acceleration, and \( \alpha \) is angular acceleration.5. **Centripetal Acceleration**: - (e) In rotational motion, centripetal acceleration (\( a_c \)) accompanies tangential acceleration. For a child on a merry-go-round, this exists with the tangential component. The equation for centripetal acceleration is \( a_c = \frac{v^2}{r} \). The force direction is towards the center, expressed as \( F_c = m a_c \) or \( F_c = \frac{mv^2}{r} \).**Diagram**:- The diagram shows a circle illustrating rotational motion with angular displacement \( \Delta \theta \) and an arc length \( \Delta s \). Vectors indicate tangential velocity and angular velocity \( \omega \) around a reference line.

Yes, the arc length between the angular positions θ1 and θ2 is given by =r(∆θ) Where, ∆θ=angular displacement in radians. a) yes it is true , If the value of ∆θ tends to zero, arc tends to become a straight line.Combination of infinite small straight lines oriented in different directions will become a curve. Part b : ∆s=r∆θIf we divide by time on both sides,∆s∆t=r∆θ∆tLinear velocity v = ∆s∆tv=r∆θ∆tω=∆θ∆tv=rω As ∆t tends to zero,the arc ∆s will be perpendicular to the radial line As time tends to zero, the arc AA' will be perpendicular to the radial line OA.Part C : Let v1 be the linear velocity when angular velocity is ω1 , v1= rω1Let v2 be the linear velocity when angular velocity is ω2 , v2= rω2v2-v1 =r(ω2-ω1)∆v=r(∆ω)The given statement hold good.It is true Note ** : We are supposed to answer only first three subparts of the question at a time, please post the remaining subparts as a separate question.