Problem: Surfer on a wave

Problem

A surfer ($m = 70\;\text{kg}$) slides down a wave. The wave force acts at $60^\circ$ from the horizontal and the acceleration is horizontal. Find the magnitude of the wave force and the surfer’s acceleration.

Solution

Acting on the surfer are the weight $\vv{P}$ (vertical, downward) and the wave force $\vv{F}_A$, tilted at $60^\circ$ from the horizontal. The acceleration is horizontal: there is no vertical acceleration. We choose $x$ horizontal (in the direction of motion) and $y$ vertical upward. Newton’s second law in components:

$\begin{cases} F_A\cos 60^\circ = m a \\ F_A\sin 60^\circ - m g = 0 \end{cases}$

Magnitude of the force. From the vertical equation, the vertical component of $\vv{F}_A$ must exactly balance the weight:

$F_A = \frac{m g}{\sin 60^\circ} = \frac{70 \cdot 9{,}81}{0{,}866}$

$\ev{F_A \approx 793\;\text{N}}$

Acceleration. From the horizontal equation, the horizontal component of $\vv{F}_A$ produces the acceleration:

$a = \frac{F_A\cos 60^\circ}{m} = \frac{793 \cdot 0{,}5}{70}$

$\ev{a \approx 5{,}66\;\text{m/s}^2}$

Links

Topics: Dynamics Concepts: Newton’s second law Skills: Vector resolution · Use of trigonometry