Problem 1.11 — Minimum angle for constant-speed trajectory on a curve

Problem Statement

A point moves along the curve $y = a\sin(px)$ at constant speed $v$. Find the acceleration and radius of curvature at the crests.

Given Information

• See problem statement for all given quantities.

Physical Concepts & Formulas

This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.

• See the step-by-step solution for the specific equations applied.
• All quantities are in SI units unless otherwise stated.

Step-by-Step Solution

Step 1 — Identify given quantities and set up the problem: Curvature formula: $\kappa = |y”|/(1+y’^2)^{3/2}$, so $R = (1+y’^2)^{3/2}/|y”|$

Step 2 — Apply the relevant physical law or equation: $y’ = ap\cos(px)$, $y” = -ap^2\sin(px)$

Step 3 — Solve algebraically for the unknown: At the crests $x = \pi/(2p)$: $y’=0$, $|y”| = ap^2$

Step 4 — Substitute numerical values with units: $$R_{\min} = \frac{1}{ap^2}$$

Step 5 — Compute and check the result: Since speed is constant, $w_\tau = 0$, and the total acceleration equals the centripetal:

Step 6: $$|\vec w| = w_n = \frac{v^2}{R} = \boxed{ap^2v^2}$$

Worked Calculation

$$R_{\min} = \frac{1}{ap^2}$$

$$|\vec w| = w_n = \frac{v^2}{R} = \boxed{ap^2v^2}$$

$$\text{Numerical result} = \text{given expression substituted with values}$$

Curvature formula: $\kappa = |y”|/(1+y’^2)^{3/2}$, so $R = (1+y’^2)^{3/2}/|y”|$

$y’ = ap\cos(px)$, $y” = -ap^2\sin(px)$

At the crests $x = \pi/(2p)$: $y’=0$, $|y”| = ap^2$

$$R_{\min} = \frac{1}{ap^2}$$

Since speed is constant, $w_\tau = 0$, and the total acceleration equals the centripetal:

$$|\vec w| = w_n = \frac{v^2}{R} = \boxed{ap^2v^2}$$

The acceleration points toward the center of curvature (downward at the crests).

Answer

$$|\vec w| = w_n = \frac{v^2}{R} = \boxed{ap^2v^2}$$

Physical Interpretation

The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.