A 50 kg skier is pulled up a frictionless ski slope that...

A $50 \mathrm{~kg}$ skier is pulled up a frictionless ski slope that makes an angle of $8.0^{\circ}$ with the horizontal by holding onto a tow rope that moves parallel to the slope. Determine the magnitude of the force of the rope on the skier at an instant when (a) the rope is moving with a constant speed of $2.0 \mathrm{~m} / \mathrm{s}$ and $(\mathrm{b})$ the rope is moving with a speed of $2.0 \mathrm{~m} / \mathrm{s}$ but that speed is increasing at a rate of $0.10 \mathrm{~m} / \mathrm{s}^{2}$.

Key Concepts

Constant Velocity vs. Accelerated Motion

In scenarios where an object moves at constant velocity, the net force acting on it is zero, meaning any applied force exactly counteracts the component of gravitational force parallel to the plane. Conversely, if the object is accelerating, the net force is not zero and must account for both overcoming gravitational pull and providing the additional force required for acceleration. This distinction is essential for correctly applying Newton's Second Law in each case.

Frictionless Surface

A frictionless surface is an idealized concept where no frictional force opposes the motion. When dealing with problems on such surfaces, only forces like gravity and any applied forces (such as tension in a rope) need to be considered in the analysis, simplifying the calculations and focusing solely on the effects of gravity and acceleration.

Free-Body Diagram

A free-body diagram is a schematic that represents all the forces acting upon an object. It is useful in isolating the object and visually identifying the different forces, such as gravitational force, normal force, and any applied forces. This diagram is foundational for setting up the equations needed to apply Newton's Second Law in both equilibrium and accelerated situations.

Decomposition of Forces on an Inclined Plane

When an object is on an inclined plane, gravitational force is resolved into two components: one perpendicular to the surface and one parallel to the surface. The parallel component is responsible for the object’s movement along the plane, and understanding this decomposition is crucial for analyzing motion and calculating the necessary force to overcome gravity or produce acceleration.

Newton's Second Law

This law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. It is fundamental in analyzing all forces acting on an object in motion or at rest, making it essential for determining the force needed to produce a given acceleration or to balance the forces when moving at constant speed.

Transcript

Please give Ace some feedback