6.3 — Non-uniform circular motion

If an object moves on a circle but its speed also changes, acceleration has two components: radial (toward center) and tangential (along motion).

Two components of acceleration

\[ \vec a = \vec a_r + \vec a_t \]

• Radial: a_r = v^2/r — toward center — responsible for changing velocity direction
• Tangential: a_t = dv/dt — tangent to path — responsible for changing velocity magnitude

Total acceleration magnitude

\[ a = \sqrt{a_r^2 + a_t^2} \]

Two components of force

Newton's second law splits similarly:

• Centripetal: F_r = mv^2/r
• Tangential: F_t = m a_t = m dv/dt

Example 1 — train speeding up in a turn

Train on a horizontal curve of radius 200 m at 20 m/s, with tangential acceleration 1 m/s².

• a_r = 2\ m/s^2 (toward center)
• a_t = 1\ m/s^2 (along motion)
• a = \sqrt{5} \approx 2.24\ m/s^2

Acceleration angle from radial direction: \arctan(1/2) = 26.6°

Example 2 — simple pendulum at various angles

Pendulum of length L released at angle \theta_0. When it swings through angle \theta:

• Radial (toward pivot): tension T and gravity component mg\cos\theta. Their difference gives the centripetal force: T - mg\cos\theta = mv^2/L
• Tangential (perpendicular to string, along motion): gravity only: -mg\sin\theta (negative because restoring)

Result: motion is a partial circle, but with varying speed — fast at bottom, slow at top.

Example 3 — cyclist in a curve

Cyclist in a curve of radius 30 m at 10 m/s decelerating at -2 m/s².

• a_r = 3.33\ m/s^2
• a_t = -2\ m/s^2
• Total: \sqrt{15.1} \approx 3.89\ m/s^2

Net force: cyclist leans into the turn so the horizontal component of weight provides centripetal — and friction handles tangential.

Example 4 — cart descending a curved surface

Cart released on a curved surface. Gravity has two components at each instant:

• Radial (toward center of curvature): combines with surface normal
• Tangential (along motion): accelerates the cart

Note: on curved paths, the normal force can be less than mg\cos\theta (due to centripetal need).

What you should be able to do

• Distinguish radial vs tangential components of acceleration
• Compute total acceleration by Pythagoras
• Identify which forces are radial and which tangential
• Apply to pendulum, cyclist, train in curves

Preview of §6.4

Non-inertial frames. Fictitious forces — centrifugal, Coriolis. Why does sand fly outward on a rotating disc?

📚 See also: Halliday Vol 1, Ch 6, §6.5.

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