A child swings a rock of mass m in a horizontal circle using...

A child swings a rock of mass $m$ in a horizontal circle using a rope of length $L$. The rope makes a constant angle $\theta$ with the horizontal. The rock moves at constant speed $v .$ What is the tension in the rope? Express the tension in terms of $m, g, v, L,$ and $\theta$

Key Concepts

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_net = m·a). In problems involving circular motion, this law is applied separately along the vertical and horizontal directions to set up equations that describe the balance of forces. This principle underpins the entire analysis of the forces experienced by the object in circular motion.

Equilibrium in the Vertical Direction

Equilibrium in the vertical direction refers to the condition where the sum of the forces acting vertically is zero, ensuring that there is no vertical acceleration. In the scenario provided, the vertical component of the tension must exactly balance the gravitational force (mg). This balance is essential for the system to remain in a steady state without vertical motion.

Vector Decomposition of Forces

Vector decomposition is the process of breaking down a force into its component vectors along chosen axes, often horizontal and vertical. This is crucial for analyzing problems like the one given, because the tension in the rope has components that balance the gravitational force vertically and provide the centripetal force horizontally. Decomposing forces allows a systematic application of Newton's laws in each independent direction.

Uniform Circular Motion

Uniform circular motion describes the motion of an object traveling at a constant speed along a circular path. In this context, even though the speed is constant, the object experiences acceleration due to the continuous change in direction. This acceleration is always directed towards the center of the circle, and its magnitude is given by v²/r, where v is the speed and r is the radius of the circle.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, and it always points towards the center of the circle. In the problem, the horizontal component of the tension provides the necessary centripetal force, which can be expressed as m·(v²/r), where m is the mass and v is the speed of the object. Understanding centripetal force is key to relating the kinematics of circular motion to the dynamics provided by the tension.

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