Course Title: Dynamics of aPage 1 of 16
Particle
Course Code: DYN 233
Plane Curvilinear Motion: Normal and
Tangential Components
Objectives
1. Establish and interpret the normal–tangential coordinate system for plane curvilinear motion.
2. Describe velocity in terms of path motion and explain why it is always tangent to the trajectory.
3. Decompose acceleration into tangential and normal components and explain the physical meaning of
each.
4. Interpret geometrically how changes in speed and changes in direction produce acceleration.
5. Apply the normal–tangential formulation to circular motion as a special case.
1 Normal and Tangential Coordinates (𝒏-𝒕)
In curvilinear motion, path variables are used, measured along the tangential 𝑡 and normal 𝑛
directions to the particle’s path. These coordinates provide a natural and convenient description because
the 𝑛- and 𝑡-coordinates move along the path with the particle (Fig 1). The positive 𝑛-direction at any
position is always taken toward the center of curvature of the path, and it shifts sides if the curvature changes
direction.
Fig 1
1.1 Base Vectors
The base vectors 𝐞𝑡 and 𝐞𝑛 , the unit tangent and the unit normal, at a point on a curved path serve
as mutually perpendicular unit vectors that form the basis for expressing velocity and acceleration. Unlike
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: or go to YT: @engora-ph stuvia
, Page 2 of 16
the fixed rectangular unit vectors, their directions are not constant but depend on the particle’s position
along the path. The vector 𝐞𝑡 is tangent to the path and points in the direction of increasing path coordinate
𝑠, whereas 𝐞𝑛 is normal to the path and directed toward the center of curvature 𝐶. As the particle moves,
these base vectors change direction while remaining unit and perpendicular.
To obtain these derivatives, the base vectors are first expressed in rectangular components:
𝒆𝑡 = − 𝑠𝑖𝑛 𝛽 𝒊 + 𝑐𝑜𝑠 𝛽 𝒋
𝒆𝑛 = − 𝑐𝑜𝑠 𝛽 𝒊 − 𝑠𝑖𝑛 𝛽 𝒋
𝑑𝐢
Differentiating with respect to time and noting that 𝑑𝑡
=
𝑑𝐣
= 0 gives
𝑑𝑡
𝐞̇ 𝑡 = (− cos 𝛽 𝐢 − sin 𝛽 𝐣)𝛽̇
𝐞̇ 𝑛 = (sin 𝛽 𝐢 − cos 𝛽)𝛽̇
Comparison with the original expressions yields the
compact relations
𝐞̇ 𝑡 = 𝛽̇ 𝐞𝑛
𝐞̇ 𝑛 = −𝛽̇ 𝐞𝑡
Thus, each base vector changes only in direction, and its Fig 2
time derivative is perpendicular to the vector itself.
1.2 Velocity
Let 𝐞𝑡 and 𝐞𝑛 be unit vectors in the tangential and normal directions, respectively (Fig 3). During
a differential motion, the particle moves a distance 𝑑𝑠 along the curve related to the radius of curvature 𝜌
by
𝑑𝑠 = 𝜌𝑑𝛽
Hence the speed is
𝑑𝑠 𝑑𝛽
𝑣= = 𝑠̇ = 𝜌
𝑑𝑡 𝑑𝑡
and the velocity vector becomes
𝐯 = 𝑣𝐞𝑡 = 𝜌𝛽̇ 𝐞𝑡 1.1
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: or go to YT: @engora-ph stuvia
, Page 3 of 16
Thus, velocity is always tangent to the path.
Fig 3
1.3 Acceleration
𝑑𝐯
Acceleration is defined as 𝐚 = . Differentiating 𝐯 = 𝑣𝐞𝑡 using the product rule gives
𝑑𝑡
𝑑𝐯 𝑑(𝑣𝐞𝑡 )
𝐚= =
𝑑𝑡 𝑑𝑡
𝐚 = 𝑣̇ 𝐞𝑡 + 𝑣𝐞𝑡̇
To complete the acceleration expression, the rate of change of the tangential unit vector must be
determined. As the particle moves from 𝐴 to 𝐴′ (Fig 4), the unit vector 𝐞𝑡 rotates through a differential
angle 𝑑𝛽. Since 𝐞𝑡 has constant magnitude, its differential change 𝑑𝐞𝑡 is due solely to this rotation, and its
magnitude equals the length of the arc generated on the unit circle. Thus,
|𝑑𝐞𝑡 | = 𝑑𝛽
The change in 𝐞𝑡 is directed toward the normal direction, so
𝑑𝐞𝑡 = 𝐞𝑛 𝑑𝛽.
Dividing by 𝑑𝛽 gives
𝑑𝐞𝑡
= 𝐞𝑛
𝑑𝛽
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: or go to YT: @engora-ph stuvia
Particle
Course Code: DYN 233
Plane Curvilinear Motion: Normal and
Tangential Components
Objectives
1. Establish and interpret the normal–tangential coordinate system for plane curvilinear motion.
2. Describe velocity in terms of path motion and explain why it is always tangent to the trajectory.
3. Decompose acceleration into tangential and normal components and explain the physical meaning of
each.
4. Interpret geometrically how changes in speed and changes in direction produce acceleration.
5. Apply the normal–tangential formulation to circular motion as a special case.
1 Normal and Tangential Coordinates (𝒏-𝒕)
In curvilinear motion, path variables are used, measured along the tangential 𝑡 and normal 𝑛
directions to the particle’s path. These coordinates provide a natural and convenient description because
the 𝑛- and 𝑡-coordinates move along the path with the particle (Fig 1). The positive 𝑛-direction at any
position is always taken toward the center of curvature of the path, and it shifts sides if the curvature changes
direction.
Fig 1
1.1 Base Vectors
The base vectors 𝐞𝑡 and 𝐞𝑛 , the unit tangent and the unit normal, at a point on a curved path serve
as mutually perpendicular unit vectors that form the basis for expressing velocity and acceleration. Unlike
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: or go to YT: @engora-ph stuvia
, Page 2 of 16
the fixed rectangular unit vectors, their directions are not constant but depend on the particle’s position
along the path. The vector 𝐞𝑡 is tangent to the path and points in the direction of increasing path coordinate
𝑠, whereas 𝐞𝑛 is normal to the path and directed toward the center of curvature 𝐶. As the particle moves,
these base vectors change direction while remaining unit and perpendicular.
To obtain these derivatives, the base vectors are first expressed in rectangular components:
𝒆𝑡 = − 𝑠𝑖𝑛 𝛽 𝒊 + 𝑐𝑜𝑠 𝛽 𝒋
𝒆𝑛 = − 𝑐𝑜𝑠 𝛽 𝒊 − 𝑠𝑖𝑛 𝛽 𝒋
𝑑𝐢
Differentiating with respect to time and noting that 𝑑𝑡
=
𝑑𝐣
= 0 gives
𝑑𝑡
𝐞̇ 𝑡 = (− cos 𝛽 𝐢 − sin 𝛽 𝐣)𝛽̇
𝐞̇ 𝑛 = (sin 𝛽 𝐢 − cos 𝛽)𝛽̇
Comparison with the original expressions yields the
compact relations
𝐞̇ 𝑡 = 𝛽̇ 𝐞𝑛
𝐞̇ 𝑛 = −𝛽̇ 𝐞𝑡
Thus, each base vector changes only in direction, and its Fig 2
time derivative is perpendicular to the vector itself.
1.2 Velocity
Let 𝐞𝑡 and 𝐞𝑛 be unit vectors in the tangential and normal directions, respectively (Fig 3). During
a differential motion, the particle moves a distance 𝑑𝑠 along the curve related to the radius of curvature 𝜌
by
𝑑𝑠 = 𝜌𝑑𝛽
Hence the speed is
𝑑𝑠 𝑑𝛽
𝑣= = 𝑠̇ = 𝜌
𝑑𝑡 𝑑𝑡
and the velocity vector becomes
𝐯 = 𝑣𝐞𝑡 = 𝜌𝛽̇ 𝐞𝑡 1.1
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: or go to YT: @engora-ph stuvia
, Page 3 of 16
Thus, velocity is always tangent to the path.
Fig 3
1.3 Acceleration
𝑑𝐯
Acceleration is defined as 𝐚 = . Differentiating 𝐯 = 𝑣𝐞𝑡 using the product rule gives
𝑑𝑡
𝑑𝐯 𝑑(𝑣𝐞𝑡 )
𝐚= =
𝑑𝑡 𝑑𝑡
𝐚 = 𝑣̇ 𝐞𝑡 + 𝑣𝐞𝑡̇
To complete the acceleration expression, the rate of change of the tangential unit vector must be
determined. As the particle moves from 𝐴 to 𝐴′ (Fig 4), the unit vector 𝐞𝑡 rotates through a differential
angle 𝑑𝛽. Since 𝐞𝑡 has constant magnitude, its differential change 𝑑𝐞𝑡 is due solely to this rotation, and its
magnitude equals the length of the arc generated on the unit circle. Thus,
|𝑑𝐞𝑡 | = 𝑑𝛽
The change in 𝐞𝑡 is directed toward the normal direction, so
𝑑𝐞𝑡 = 𝐞𝑛 𝑑𝛽.
Dividing by 𝑑𝛽 gives
𝑑𝐞𝑡
= 𝐞𝑛
𝑑𝛽
© 2026 Prepared by ENGORA—the library of engineering notes. Redistribution, reuploading, resale, or public posting without
permission is not allowed. For more of these, contact: or go to YT: @engora-ph stuvia
