AP Physics 1 · Unit 5 · Topic 5.2
Connecting Linear and Rotational Motion
Every point fixed in a rotating rigid system shares the same angular displacement, angular velocity, and angular acceleration. Its linear motion, however, depends on how far that point lies from the rotation axis.
Learning Goals
After this lesson, you should be able to:
- derive arc length from angular displacement in radians;
- connect tangential speed with angular speed;
- connect tangential acceleration with angular acceleration;
- compare points at different radii on one rigid system;
- distinguish tangential and centripetal acceleration;
- translate angular graphs into linear graphs for a chosen radius;
- apply the relations to rims, belts, cables, and rotating platforms; and
- design a video experiment testing proportionality with radius.
1. The Three Bridge Equations
For a point fixed at perpendicular distance \(r\) from a fixed rotation axis,
Angles must be in radians. The subscript \(t\) means tangential: the linear direction is tangent to the circular path at that instant. These equations relate a particular point's linear motion to the rigid system's angular motion.
2. Same Angle, Different Linear Motion
3. Why Radians Make \(s=r\theta\) Work
A radian is defined by the ratio of arc length to radius:
The ratio is dimensionless, which is why radians fit directly into the equation. Substituting degrees without conversion gives an incorrect scale.
Example: Through \(2.4\,\mathrm{rad}\), points at \(0.15\,\mathrm m\) and \(0.45\,\mathrm m\) travel
The outer point travels three times as far because its radius is three times as large.
4. Deriving Tangential Speed
For a point whose radius is constant, divide the arc relation by the elapsed time:
The velocity vector is tangent to the circle and changes direction continuously. \(v_t=r\omega\) gives its magnitude with a sign convention along the tangent; it does not mean the vector points in the angular-velocity direction.
5. Worked Example: Speeds on a Rotating Platform
A disk rotates at \(18\,\mathrm{rad/s}\). Find the tangential speeds of points at \(r_A=0.080\,\mathrm m\) and \(r_B=0.32\,\mathrm m\).
Answer: The outer point moves four times faster linearly even though both points complete each revolution in the same time and have the same angular speed.
6. Deriving Tangential Acceleration
Differentiate or compare changes in \(v_t=r\omega\) while radius remains fixed:
Tangential acceleration changes the magnitude of the tangential velocity. If \(\omega\) and \(\alpha\) have the same rotational sign, the point speeds up; if their signs differ, it slows down.
7. Worked Example: A Disk Slows Down
A disk has angular acceleration \(-6.0\,\mathrm{rad/s^2}\). At a point \(0.25\,\mathrm m\) from the axis,
Answer: The signed tangential acceleration is \(-1.5\,\mathrm{m/s^2}\) relative to the chosen positive tangent. A point at \(0.10\,\mathrm m\) would have only \(-0.60\,\mathrm{m/s^2}\), although both share \(\alpha=-6.0\,\mathrm{rad/s^2}\).
8. What Every Point Shares—and What It Does Not
| Quantity | Same for all points on one rigid system? | Radius dependence |
|---|---|---|
| \(\Delta\theta\) | Yes | None |
| \(\omega\) | Yes | None |
| \(\alpha\) | Yes | None |
| \(\Delta s\) | No | \(\propto r\) |
| \(v_t\) | No | \(\propto r\) |
| \(a_t\) | No | \(\propto r\) |
At the axis, \(r=0\), so arc length, tangential speed, and tangential acceleration are zero even while the rigid system has nonzero angular quantities.
9. Tangential and Centripetal Acceleration
A rotating point can have two perpendicular acceleration components:
Even when \(\alpha=0\), uniform circular motion has \(a_t=0\) but generally \(a_c\ne0\). Centripetal acceleration always points toward the axis; tangential acceleration lies along or opposite the instantaneous velocity.
10. Total Linear Acceleration
Because tangential and centripetal components are perpendicular,
Example: At \(r=0.20\,\mathrm m\), a wheel has \(\omega=10\,\mathrm{rad/s}\) and \(\alpha=4.0\,\mathrm{rad/s^2}\):
The nearly radial total does not make the smaller tangential component zero.
11. Translating Graphs with Radius
For a chosen fixed radius, multiply every vertical value of an angular graph by \(r\):
The time coordinates, zeros, sign changes, and overall graph shapes remain the same. A point at twice the radius has twice the vertical scale on all three corresponding linear graphs. This provides a quick proportional-reasoning method without repeating a full calculation at every time.
12. Worked Graph Scaling
An \(\omega(t)\) graph rises linearly from \(2.0\) to \(8.0\,\mathrm{rad/s}\) over \(3.0\,\mathrm s\). For a point at \(r=0.40\,\mathrm m\), the tangential-speed graph rises from
to
Its slope is \(a_t=(3.2-0.80)/3.0=0.80\,\mathrm{m/s^2}\). The angular slope is \(\alpha=(8.0-2.0)/3.0=2.0\,\mathrm{rad/s^2}\), confirming \(a_t=r\alpha\).
13. Rim, Belt, and Cable Models
If a taut belt or cable moves with a rim without slipping, its linear speed equals the rim's tangential speed:
Example: A \(0.12\,\mathrm m\)-radius pulley rotates at \(15\,\mathrm{rad/s}\), so a no-slip cable moves at \(1.8\,\mathrm{m/s}\). If slipping occurs, this equality is not valid. Always state the contact assumption rather than applying the equation automatically.
14. Investigation: Test Linear Scaling with Radius
Use a turntable or wheel, two visible markers at different radii, a ruler, and overhead video.
- Measure each marker's distance from the axis and state the positive direction.
- Record several revolutions with the camera centered over the axis.
- Track angular position and each marker's arc position at equal times.
- Plot \(s\) against \(\theta\) for each marker; compare best-fit slopes with measured radii.
- Estimate \(\omega\) and compare \(v_t/r\) for the two markers.
- During a speed-changing interval, compare \(a_t/r\) with the shared \(\alpha\).
- Discuss parallax, axis location, marker width, frame rate, radius uncertainty, and disk wobble.
Evidence supports the model when \(s\)-versus-\(\theta\) is linear through the origin and its slope agrees with \(r\) within uncertainty.
15. Common Traps and Key Takeaways
- Use radians in \(s=r\theta\).
- All rigidly attached points share angular quantities, not linear quantities.
- The outer point has greater \(s\), \(v_t\), and \(a_t\) for the same angular motion.
- Tangential velocity is perpendicular to the radius, not directed outward.
- \(a_t\) changes speed; \(a_c\) changes direction.
- Constant angular speed does not mean zero total acceleration.
- State a no-slip assumption before matching a belt or cable speed to rim speed.
- Do not mix diameter and radius.
A rigid disk starts with \(\omega_i=3.0\,\mathrm{rad/s}\) and has constant \(\alpha=2.0\,\mathrm{rad/s^2}\) for \(4.0\,\mathrm s\). Points A and B lie at \(0.10\,\mathrm m\) and \(0.35\,\mathrm m\) from the axis. Find the shared angular displacement and final angular speed; then find each point's arc length, final tangential speed, tangential acceleration, and final centripetal acceleration. Sketch corresponding \(\omega(t)\), \(v_A(t)\), and \(v_B(t)\) graphs and explain which features remain identical.
Concept scope cross-checked with the College Board AP Physics 1 Course and Exam Description and the open educational references OpenStax University Physics: Relating Angular and Translational Quantities and OpenStax College Physics for AP Courses: Angular Acceleration. Explanations, diagram, examples, graph task, checkpoint, and investigation are independently written for Insomnia Studyroom.
Official curriculum reference: College Board AP Physics 1 course page. The explanation and worked example are independently written for this study site.