Circular Motion in Applied Math defines the principles that guide the movement of particles in a circle. Suppose a particle moves along a circular path (called an arc) from a point A to a point B covering an angle ᶿ within a time interval t;
- Angular velocity (ω) = angle covered (ᶿ)/ time interval (t), where ᶿ is in radians.
- A radian = arc length (s)/ radius (r).
- For 1 revolution (360o), radian = 2πr/ r = 2π.
- Time for 1 rev = T, so ω = 2π/ T; T = 2π/ ω.
- Frequency (f) = 1 rev/ T i.e. 1/ (2π/ ω) = ω/ 2π.
- A particle moving in a circular path moves with a constant velocity (v) in Circular Motion in Applied Math. If v is not constant, ω = ∆ᶿ/ ∆t. Engineering in Kenya has more information.
For Circular Motion in Applied Math, consider a circle of radius r, and a particle moving from point A to point P covering an angle ᶿ;
- V = arc length/ t.
- Arc length = radius × angle = rᶿ.
- Therefore, v = r (ᶿ/ t)
- But ᶿ/ t = ω; so v = rω.
Consider the same arc as in above with arc AB being s and name it triangle 1;
Also substitute the arc s with a straight line from A to B and name it triangle 2;
For Circular Motion in Applied Math, we find that triangle 1 and triangle 2 are similar, since they have two similar sides and a common angle. Using triangle 2 with the similar sides being v1 and v2 respectively and line AB being ∆v;
- v1 = v2 = v
- Arc length (s)/ r (triangle 1) = ∆v/ v (triangle 2), i.e. ∆s/ r = ∆v/ v.
- So, ∆v = (v/ r) ∆s.
- Divide both sides by ∆t; ∆v/ ∆t = (v/ r) (∆s/ ∆t).
- (∆v/ ∆t) = rate of change of velocity, which is called acceleration (a) in Circular Motion in Applied Math.
- (∆s/ ∆t) = rate of change of displacement, which is called velocity in Circular Motion in Applied Math.
- So, a = (v/ r) v = v2/ r.
- But v = rω, so a = (rω)2/ r = ω2r; called angular acceleration in Circular Motion in Applied Math.
Centripetal Force in Circular Motion in Applied Math
When a body is rotated, centripetal force tends to pull the body towards the centre. Centrifugal force is antagonistic to the centripetal force. Centripetal force in Circular Motion in Applied Math acts such that the acceleration (a) is towards the centre of the rotation.
- Since F = ma, and a = ω2r = v2/ r;
- F = mω2r or F = mv2/ r. The two formulae define centripetal force in Circular Motion in Applied Math.
Motion in a Vertical Circle in Circular Motion in Applied Math
Consider a stone tied on a string rotating in a vertical circle; this motion is affected by gravity. The string experiences different tensions in this movement in Circular Motion in Applied Math.
- At the topmost of the rotation, since the weight of the stone is acting downwards, it reduces the tension on the string, therefore tension (T) = (mv2/ r) – (mg).
- At a point perpendicular to the line of action of gravity, the stone has no vertical component and thus tension (T) = mv2/ r.
- At the bottom of the rotation, since the weight of the stone is acting downwards, it adds to the tension on the string, therefore tension (T) = (mv2/ r) + mg.
Vehicles Negotiating Bends in Circular Motion in Applied Math
Centripetal force required to force a car go round a bend has to be provided by the frictional force in Circular Motion in Applied Math. Assuming that there is no skidding, the maximum safe speed is governed by the limiting friction.
- So, centripetal force = frictional force (for maximum safe speed).
- i.e. mv2/ r = µR, where µ is the coefficient of friction.
- But R = mg, since action and reaction forces are equal and opposite. So, friction = µmg.
- Divide both sides by m; v2/ r = µg, v2 = rµg.
- So for maximum safe speed, v = √rµg.
Conclusion on Circular Motion in Applied Math
Reliance on friction is dangerous, as one cannot always calculate the maximum safe speed. This can be reduced by road banking. Banking in Circular Motion in Applied Math is when the road is tilted slightly to reduce the reliance on friction. Given the angle of banking is ᶿ, and the radius of the bend is r, the maximum safe speed v = √gr tanᶿ in Circular Motion in Applied Math.