centripetal acceleration and tangential acceleration
this acceleration arises due to change in speed of particle or change in magnitude of velocity hence if particle velocity magnitude is changing due to this a acceleration will come in picture which is called tangential acceleration this is also a vector quantity hence its will have definitely direction now direction depend upon the velocity magnitude if velocity magnitude is increasing then direction will be along the velocity direction and if velocity is decreasing then direction will be opposite to velocity direction tangent to the radius as shown in above picture now its value is define as a𝓉 = d|v|/dt remember this is change in speed upon change in time concept it is not change in velocity upon change time v = velocity and | v | = speed hence in tangential acceleration has a condition if velocity is constant then d|v|/dt = 0 because constant differentiation is zero so in this case a𝓉 = d|v|/dt =0 so only centripetal acceleration will be there no tangential acceleration sometime centripetal acceleration is also called normal acceleration or perpendicular acceleration because it is perpendicular to the surface and tangential acceleration is called parallel acceleration because it is parallel to surface as shown above figure .
Net acceleration(a𝓷𝓮𝓽) this is due to total change in velocity due to direction and velocity magnitude both it is called net acceleration now its value is a𝓷𝓮𝓽 = |dv/dt| now its direction will not be in velocity direction because a𝒸 and a𝓉 are always perpendicular hence a𝓷𝓮𝓽 resultant will be a𝓷𝓮𝓽 = √a𝒸²+a𝓉² now for direction use triangle law tan𝝰 = a𝒸/a𝓉 now we will see some questions to clear concepts.
Q if tangential velocity v= 2t and radius r =9m find a𝒸,a𝓉 and a𝓷𝓮𝓽 at t =3s.
Ans for velocity v= 2t put t=3 v =2*3 =6m/s
centripetal acceleration a𝒸 = v²/r = 6*6/9 = 4m/s²
tangential acceleration a𝓉 = d| v |/dt = 2m/s²
net acceleration a𝓷𝓮𝓽 = √a𝒸²+a𝓉² = √4²+2² = √20
Q2 if 𝛚 = 2t and radius r =2 find a𝒸,a𝓉 and a𝓷𝓮𝓽
now you can solve this problem yourself using v = r𝛚 .
Angular acceleration(𝛂 ) change in angular velocity divided by change in time 𝛂avg = ∆𝛚/∆t or 𝛂inst = d𝛚/dt
Q if 𝛚 = t² +1 find 𝛂 in 0 to 2s and at 2s.
Ans angular acceleration in 0 to 2s put t=0 𝛚0 = 1 now put t=2s 𝛚2 = 2*2+1 = 5 ∆𝛚 = 5-1 = 4 hence 𝛂avg = ∆𝛚/∆t = 4/2 =2 rad/s²
now for 𝛂 at 2s 𝛂inst = d𝛚/dt = 2t = 2*2 = 4 rad/s²
now one relation in angular acceleration and linear acceleration
a𝓉 = r𝛂 where a𝓉 is tangential acceleration .
Q if | v | = constant in a circular motion then find 𝛂, a𝓉,a𝒸 which will be constant ?
Ans since | v | is constant then 𝛚 = v/r here both v and r constant then 𝛚 will be constant hence 𝛂 = dw/dt = 0 which is constant now for a𝓉 we know a𝓉 = r𝛂 = 0 which is also constant now for a𝒸 = v²/r here v and r are constant then a𝒸 magnitude will be constant but a𝒸 is a vector quantity and its direction is variable all the time in circular motion hence it will not be constant it will be variable remember this is important concept this is all about today topic we will continue in next post i hope you have enjoyed learning centripetal acceleration and tangential acceleration thanks for reading.
dated 16th sep 2018