Question

A particle moves in a plane according to X=Rsinwt + wRt Y=Rcoswt+R where w and R constant this curved called cycloid path traced out by a point on rim of wheel that slipping with x-axis sketch the path calculate intantaneous velocity and acceleration when particle is its axis maximun and minimun value of Y

Answer 1

maximum point y= 2R

vₓ = wR , v_{y} = 0, aₓ = 0, a{y} = - Rw²

minimum point y=R

vₓ = wR , v_{y} = - R w, aₓ = - R w², a_{y} = 0

Step-by-step explanation:

The definition of velocity is

v = dr / dt

vₓ = dx / dt

`v_(y)` = dy / dt

vₓ = Rw cos wt + wR

v_{y} = -Rw sin wt

acceleration is defined by

a = dv / dt

aₓ = -Rw² sin wt

a_{y} = - R w² cos wt

these are the general expressions for velocity and acceleration, to find the explicit values ​​for the maximum and minimum y points, let's find these points and substitute

maximum point

y = R cos wt + R

the heat is maximum when the cosine is worth 1

y_max = 2R

at this point the speed is

vₓ = wR

v_{y} = 0

the acceleration is

aₓ = 0

a_{y} = - Rw²

minimum point

this occurs when the cosine is zero

y = R

speed is

vₓ = wR

v_{y} = - R w

acceleration is

aₓ = - R w²

a_{y} = 0