Lecture 5, Cylindrical Coordinate Frames
1. Cylindrical coordinates for 3D motion
z
⃗𝒛
𝒆
⃗𝑡
𝒆
P
⃗𝑟
𝒆
z
O
𝜃
x
y
r
⃗𝒆𝑡
𝑃′
⃗𝑟
𝒆
In the case of 3D motion, the position of a particle P can be described by cylindrical
coordinates (𝑟, 𝜃, 𝑧). The position vector is written as:
⃗⃗⃗⃗⃗ = 𝑟𝒆
⃗ 𝑟 + 𝑧𝒆
⃗𝑧
𝑂𝑃
⃗ 𝑟 still follows the 2D polar frame
The projection of ⃗⃗⃗⃗⃗
𝑂𝑃 on the x-y plane, ⃗⃗⃗⃗⃗⃗⃗
𝑂𝑃′ = 𝑟𝒆
⃗ 𝑧 , orthogonal to both 𝒆
⃗ 𝑟 and 𝒆
⃗ 𝑡 does not change. Therefore,
expressions. The unit vector 𝒆
in deriving velocity and acceleration by vector differentiation of position vector ⃗⃗⃗⃗⃗
𝑂𝑃, no
⃗ 𝑧.
need to include 𝒆
The velocity and acceleration of the motion have an extra term in the z axis.
⃗ 𝑟 + 𝑟𝜃̇𝒆
⃗ 𝑡 + 𝑧̇ 𝒆
⃗𝑧
𝒗 = 𝑟̇ 𝒆
⃗𝑧
𝒂 = (𝑟̈ − 𝑟𝜃̇ 2 )𝒆𝑟 + 𝑟𝜃̈𝒆𝑡 + 2𝑟̇ 𝜃̇𝒆𝑡 + 𝑧̈ 𝒆
2. Example
1) A particle moves in a helical path given by the equation
⃗ = 𝒊 𝑏 sin 𝜔𝑡 + ⃗𝒋 𝑏 cos 𝜔𝑡 + ⃗⃗⃗𝒌 𝑐𝑡 2 ,
𝒓
show that the magnitude of the acceleration of the particle is constant, provided b,
𝜔, and c are constant.
The position vector can be written as:
⃗ = 𝑏𝒆
⃗ 𝑟 + 𝑐𝑡 2 𝒆
⃗𝑧
𝒓
𝑑
⃗ 𝑟 = 𝜔𝒆
⃗ 𝑡 . Use the velocity and acceleration equations in the
Where, 𝑑𝑡 𝒆
Cylindrical frame:
⃗ 𝑟 + 𝑟𝜃̇𝒆
⃗ 𝑡 + 𝑧̇ 𝒆
⃗𝑧
𝒗 = 𝑟̇ 𝒆
⃗𝑧
𝒂 = (𝑟̈ − 𝑟𝜃̇ 2 )𝒆𝑟 + 𝑟𝜃̈𝒆𝑡 + 2𝑟̇ 𝜃̇𝒆𝑡 + 𝑧̈ 𝒆
It is clear that the radial velocity is zero, the transverse velocity is 𝑏𝜔, and
velocity in the z axis is 2𝑐𝑡, therefore, the velocity is given by:
⃗ 𝑡 + 2𝑐𝑡𝒆
⃗𝑧
𝒗 = 𝑏𝜔𝒆
Acceleration is given by:
⃗ 𝑟 + 2𝑐𝒆
⃗𝑧
𝒂 = −𝑏𝜔2 𝒆
The magnitude of acceleration is:
|𝒂| = √𝑏 2 𝜔 4 + 4𝑐 2
2) The ball running down the helical ramp with a constant speed of 3m/s. The ramp
descends a vertical distance of 1.5m for every full revolution. Determine the
magnitude of its acceleration.
The ramp is at an angle of 𝜑 from the horizontal:
1.5
2𝜋(1)
⃗ 𝑡 and 𝒗
⃗ 𝑧 (note that 𝒗
⃗ 𝒓 is zero),
This determines the velocity components 𝒗
̇
which in turn will determine the 𝜃 and 𝑧̇ .
𝜑 = 𝑡𝑎𝑛−1
⃗ 𝑡 = 𝑟𝜃̇𝒆
⃗ 𝑡 , and 𝒗
⃗ 𝑧 = 𝑧̇ 𝒆
⃗𝑧
𝒗
When 𝜃̇ is known, acceleration is given by (note that, 𝑟̈ = 0, 𝜃̈ = 0, 𝑧̈ = 0):
⃗𝑧
𝒂 = (𝑟̈ − 𝑟𝜃̇ 2 )𝒆𝑟 + 𝑟𝜃̈𝒆𝑡 + 2𝑟̇ 𝜃̇𝒆𝑡 + 𝑧̈ 𝒆
𝒂𝒓 = ,
𝒂𝒕 = 𝟎
𝒂𝒛 = 𝟎