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 rock moves at constant speed $v$. (a) Ignoring gravity, find the tension in the rope. (b) Now include gravity (the weight of the rock is no longer negligible, although the weight of the rope still is negligible). What is the tension in the rope? Express the tension in terms of $m, g, v, L,$ and the angle $\theta$ that the rope makes with the horizontal.

A child swings a rock of mass $m$ in a horizontal circle using a rope of length $L$. The rock moves at constant speed $v$. (a) Ignoring gravity, find the tension in the rope.
(b) Now include gravity (the weight of the rock is no longer negligible, although the weight of the rope still is negligible). What is the tension in the rope? Express the tension in terms of $m, g, v, L,$ and the angle $\theta$ that the rope makes with the horizontal.

Key Concepts

Newton's Second Law

Newton's Second Law establishes the relationship between the net force acting on a body and its acceleration, given by F = ma. This law underpins the analysis of forces in both linear and circular motion by linking the sum of all forces with the resulting motion of the object.

Force Decomposition

Force decomposition is the method of resolving a force into its perpendicular components, typically along horizontal and vertical axes. This is crucial in analyzing problems where forces act at angles, such as when a rope is inclined and both gravity and centripetal requirements affect the net force.

Gravitational Force

Gravitational force is the force exerted by the Earth on objects near its surface, often referred to as weight (mg). In scenarios involving inclined motion or non-negligible mass under gravity, this force must be considered and resolved along appropriate directions to accurately analyze the dynamics.

Centripetal Force

Centripetal force is the net force that acts inward toward the center of a circular path, required to keep an object in uniform circular motion. It is mathematically expressed as mv²/r, where m is the object’s mass, v is its speed, and r is the radius of the circle.

Uniform Circular Motion

Uniform circular motion describes the motion of an object traveling at a constant speed along a circular path. It necessitates a continuous inward (centripetal) force that keeps changing the direction of the velocity vector, maintaining the circular trajectory even though the speed remains constant.

Tension in a Rope

Tension is the force transmitted along a rope, string, or cable when forces are applied at its ends. In the context of circular motion, the tension in the rope not only provides the necessary centripetal force to maintain the motion but may also have components in other directions, especially when external forces like gravity are considered.

Transcript

00:01 In this problem, we're going to talk about circular motion.

00:03 So consider that we have a circular trajectory that has a radius r, and our goal is to describe this trajectory from a dynamical point of view.

00:15 So let's talk about the acceleration of a particle at this point that i marked in red here.

00:23 The acceleration can be decomposed into two components.

00:28 Some of them is the centripetal acceleration that is equal to b squared over r.

00:35 V here is the linear velocity and accounts for the change in direction of the trajectory.

00:41 The tangential acceleration accounts for the change in speed of the trajectory and is equal to the derivative of the speed with respect to time.

00:51 Okay, so in our problem, we have a child that's swinging a rock like this.

00:58 So notice that the circle made by the rock is a horizontal circle.

01:09 Okay.

01:12 And our goal in question a is to calculate what is the tension on the rock, on the rope, i'm sorry, ignoring gravity.

01:25 So notice that if we ignore gravity, the only force that acts on the rock, considering that this here is the rock, is the tension.

01:34 In that case, the rock cannot be, cannot make any angle with horizontal.

01:46 Actually, the rock will make an angle theta equal to zero with the horizontal.

01:53 That would be necessary because otherwise there would be no force to counterbalance the tension in the y direction...

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