(a) Calculate the acceleration of a skier heading down a 10.0^∘ slope, assuming the coefficient

step 1 If the angle is theta, then the weight MG can be decomposed into MG cosine theta perpendicular t

(a) Calculate the acceleration of a skier heading down a...

(a) Calculate the acceleration of a skier heading down a $10.0^{\circ}$ slope, assuming the coefficient of friction for waxed wood on wet snow. (b) Find the angle of the slope down which this skier could coast at a constant velocity. You can neglect air resistance in both parts, and you will find the result of Exercise 5.9 to be useful. Explicitly show how you follow the steps in the Problem-Solving Strategies.

Key Concepts

Equilibrium Condition for Constant Velocity

When an object coasts at constant velocity, its acceleration is zero, indicating that the net force acting on it is zero. For the skier, this means that the component of gravitational force down the slope (mg sin ?) must be exactly balanced by the frictional force (? mg cos ?). This equilibrium condition is essential for solving for the slope angle at which the skier can coast without accelerating.

Inclined Plane Dynamics

This concept involves resolving the gravitational force into components along and perpendicular to the slope. For an object on an inclined plane, the component of gravitational force that causes acceleration down the slope is given by mg sin ?, while the normal force is mg cos ?, which is crucial for determining the frictional force.

Frictional Force

Friction is the resistive force that opposes the motion of objects in contact. In this scenario, the frictional force is calculated as the product of the coefficient of friction and the normal force (? mg cos ?). It plays a key role in both reducing acceleration when the skier is speeding up and in creating a condition for constant velocity when it exactly balances the downhill gravitational component.

Newton's Second Law

This fundamental principle states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. It provides the framework for analyzing the forces acting on the skier and determining the acceleration down the slope by balancing the gravitational component along the incline and the opposing frictional force.

Free-Body Diagram

A free-body diagram is a visual representation that isolates an object and shows all the forces acting on it. In the context of an inclined plane, this diagram helps in breaking the gravitational force into its parallel and perpendicular components to the surface, and in clearly identifying friction as a force that opposes the motion.

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