Question

problem 8: a skier skis down the mountain. if the skiers acceleration is 3.7 m/s², find the angle of the slope. make sure to draw the free body diagram and the correct reference frame. neglect frictional type forces.

Explanation:

Step1: Analyze forces on skier

The only force acting on the skier along the slope (neglecting friction) is the component of the gravitational force, $F = mg\sin\theta$, where $m$ is the mass of the skier, $g = 9.8\ m/s^{2}$ is the acceleration - due - to - gravity, and $\theta$ is the angle of the slope. According to Newton's second law $F = ma$, where $a$ is the acceleration of the skier.

Step2: Equate the two force expressions

We set $mg\sin\theta=ma$. The mass $m$ cancels out on both sides of the equation, giving us $\sin\theta=\frac{a}{g}$.

Step3: Solve for the angle

We know that $a = 3.7\ m/s^{2}$ and $g = 9.8\ m/s^{2}$. So, $\theta=\sin^{- 1}(\frac{a}{g})=\sin^{-1}(\frac{3.7}{9.8})$.
Calculating $\sin^{-1}(\frac{3.7}{9.8})\approx22.2^{\circ}$.

Answer:

The angle of the slope is approximately $22.2^{\circ}$