A conical pendulum is formed by attaching a 53-g pebble to a 1.4 -m string. The pebble swings ar

A conical pendulum is formed by attaching a 53-g pebble to a...

A conical pendulum is formed by attaching a $53-\mathrm{g}$ pebble to a $1.4$ -m string. The pebble swings around in a circle of radius $25 \mathrm{~cm} .(a)$ What is the speed of the pebble? $(b)$ What is its acceleration? ( $c$ ) What is the tension in the string?

Key Concepts

Conical Pendulum Dynamics

A conical pendulum refers to a pendulum in which the bob moves in a horizontal circular path while the string traces out a cone. The dynamics of a conical pendulum involve both the principles of circular motion and the equilibrium of forces. The geometry of the pendulum, including the relation between the string length and the radius of the circular path (r = L sin?), plays a crucial role in determining the tangential speed, acceleration, and tension in the string.

Tension and Force Decomposition

In systems like a conical pendulum, the tension in the string is resolved into two components: one that counteracts the gravitational force (vertical component) and one that provides the centripetal force (horizontal component) required for circular motion. The vertical component typically balances the force of gravity, while the horizontal component is responsible for keeping the object in circular motion. By decomposing the tension into these components, one can set up equations that link the tension, gravitational force, and centripetal force.

Uniform Circular Motion

Uniform circular motion describes the motion of an object moving at a constant speed along a circular path. In such motion, although the speed is constant, the object is continually changing direction, which results in an acceleration directed toward the center of the circle. This concept is fundamental to analyzing any system where an object moves in a circle, as it leads to the need to consider centripetal acceleration and forces.

Centripetal Acceleration

Centripetal acceleration is the acceleration that keeps an object moving in a circular path and is directed towards the center of the circle. It is given by the formula a = v²/r, where v is the speed of the object and r is the radius of the circular path. This acceleration is critical when analyzing problems involving circular motion because it explains how a constant change in direction is achieved.

Transcript

00:01 So we're going to be looking at problem 35 from the chapter 5 of the physics 5th.

00:06 The question says, a conical pendulum is formed by attaching a 53 gram pebble to a 1 .4 meter string.

00:12 The pebble swings around in a circle of radius 25 centimetres.

00:16 For part a, what is the speed of the pebble? for part b, what is its acceleration? and for part c, what is the tension in the string? so this is the sort of basic setup we have.

00:24 We have some string length of 140 centimetres, and there's some pebble on the end of this string.

00:30 And the pebble swings around in a circle at this point forming a sort of cone shape.

00:37 And we know the radius of this cone shape is 25 centimeters.

00:41 The string also makes some angle theta with the vertical axis.

00:46 So what do we have in terms of forces? well, we have some centripetal acceleration f is equal to mb squared of r going to the right when our string is pointed to the left.

00:56 And we have some weight downwards.

00:59 Equals to mg.

01:00 And we also have some tension.

01:02 And this tension using the fact that using this angle here that we defined earlier, the vertical component of the tension is t cos theta and the horizontal is t sine theta.

01:14 So in order to calculate theta, we're just using this diagram over here...

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