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pendulum

Help me visualise the forces found in a pendulum. I know weight can be decomposed into $F_x$ and $F_y$.

Centripetal force, acceleration and velocity are three words that I'm confused. Centripetal force is in L and is equal to acceleration ? it's the gravity that move the pendulum, the weight, but there is a rope, so it's not the same situation that happens to a satellite.

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2 Answers 2

In the picture you presented weight (mg) is decomposed into $F_t$, that is tangential force, and $F_r$, that is the radial force (along the radius). Problems from rotational dynamics are usually solved in either laboratory (fixed to earth) or inertial (fixed to the rotating mass) reference frame.

Consider inertial reference frame. Forces acting on the mass at the end of the rope are weight and rope tension.

Pendulum has a velocity that is tangential, because it is moving around the circle.

Its acceleration can be considered to have both tangential and radial components. Tangential component says that the pendulum is accelerating (so the value of the velocity is changing). Radial component says that the direction of velocity is changing (because the body is moving around the circle, for instance for $\theta=90 \deg$ velocity vector points down and for $\theta=0 \deg$ is points to the left).

Centipetal force is the force that is responsible for radial acceleration. Its value is equal to $mV^2/R$ and is a sum of the rope tension and $F_r$

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""Its acceleration can be considered to have both tangential and radial components. "" Acceleration in radial direction? Rethink that, please. –  Georg Apr 12 '11 at 9:03

Hopefully you know that velocity is how fast an object's position changes, and acceleration is how fast an object's velocity changes. Both of these are vectors, so you can split them up into components along any set of perpendicular axes. For example, you could split them up into $x$ and $y$ components, or into tangential and radial components.

The same is true of the force - or rather, forces. There are multiple forces acting on the pendulum bob, both its weight and a force from the string (tension). Each of those forces is a vector, so you can split it up into components along any set of perpendicular axes.

You can also add up all the forces (as vectors) and produce something called the net force (or total force). This net force is related to the acceleration by Newton's second law,

$$\vec{F}_\text{net} = m\vec{a}$$

Again, since the net force is a vector, you can split it up into components along any set of perpendicular axes.

Now, suppose you choose one axis to point toward the center of the circle. The component of the net force that points along that axis is called the centripetal force. The component of the acceleration that points along that axis is called the centripetal acceleration. And so on.

Having chosen one axis to point toward the center of the circle, the only way you can choose the other axis to be perpendicular is to have it point parallel to the edge of the circle. The component of, say, the acceleration along that axis is called the tangential acceleration. Similarly for velocity, force, etc.

Once you get out of introductory physics it's common to replace the centripetal axis (the one that points toward the center) with an axis that points in the exact opposite direction, away from the center. The component of the net force that points along this axis could be called the radial force (or just "radial component of the net force"). It's the negative of the centripetal force, because you switched the direction of the axis. And of course, the same goes for acceleration, velocity, etc.

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