A child of mass m=50 kg sits at the end of a rope of length...

A child of mass $m=50 \mathrm{kg}$ sits at the end of a rope of length $L=3.2 \mathrm{m}$. The other end of the rope is fastened to a ceiling in a gymnasium, and the child travels so that he moves in a horizontal circle of radius $r=2.50 \mathrm{m}$ as sketched in Figure $\mathrm{P} 5.28$ a. Draw a picture showing the motion of the child. Give both a top view and a side view. b. What are all the forces on the child? Add them to your pictures in part (a). Then draw a freebody diagram for the child. c. Find the vertical and horizontal components of all the forces on the child. d. Apply Newton's second law in both the vertical ( $y$ ) and the horizontal directions. What is the acceleration along $y$ ? Find the tension in the rope.

A child of mass $m=50 \mathrm{kg}$ sits at the end of a rope of length $L=3.2 \mathrm{m}$. The other end of the rope is fastened to a ceiling in a gymnasium, and the child travels so that he moves in a horizontal circle
of radius $r=2.50 \mathrm{m}$ as sketched in Figure $\mathrm{P} 5.28$
a. Draw a picture showing the motion of the child. Give both a top view and a side view.
b. What are all the forces on the child? Add them to your pictures in part (a). Then draw a freebody diagram for the child.
c. Find the vertical and horizontal components of all the forces on the child.
d. Apply Newton's second law in both the vertical ( $y$ ) and the horizontal directions. What is the acceleration along $y$ ? Find the tension in the rope.

Key Concepts

Uniform Circular Motion

This concept describes the motion of an object traveling at a constant speed along a circular path. In such motion, although the speed is constant, the direction of the velocity vector changes continuously, resulting in an inward (centripetal) acceleration that keeps the object moving in a circle.

Free-Body Diagram

A free-body diagram is a simplified representation of an object isolated from its surroundings, showing all the external forces acting upon it. This tool is essential for visualizing forces such as gravity, tension, and normal forces, and for setting up the equations of motion based on Newton's laws.

Force Decomposition

Force decomposition involves breaking a force into its perpendicular components, typically horizontal and vertical components. This is crucial when dealing with forces at an angle, such as the tension in a rope, to analyze the separate effects each component has on the motion of the body.

Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F=ma). This principle is applied independently in the vertical and horizontal directions to quantify forces and accelerations acting on the body.

Trigonometric Relationships in Force Analysis

Trigonometric relationships, particularly sine and cosine functions, are used to relate the components of angled forces to the force’s magnitude. For instance, when analyzing the tension in a rope, these relationships help determine its horizontal component responsible for centripetal acceleration and its vertical component which balances gravity.

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