# Tangential Acceleration of pendulum

#### blackboy

Ok let's say there is a pendulum and you start 45 degrees from the left and let go. I understand the farthest down point on the circle has the greatest velocity, but I don't understand why there is no tangential acceleration. Any help?

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#### rock.freak667

Homework Helper
You'd only get a tangential acceleration if the pendulum's velocity changes with time.

$$a_t=\frac{dv}{dt}$$

If the angular velocity is constant, then so is the velocity and hence there will be no tangential acceleration.

EDIT: read dx's post...I thought of the pendulum situation incorrectly. Sorry.

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#### blackboy

How is the angular velocity constant?

#### dx

Homework Helper
Gold Member
Angular velocity is not constant. At the lowest point, gravity is pointing downward, and tension is pointing upward, and they balance, so the net force is zero. In particular, the net force in the tangential direction is zero, so there is no tangential acceleration. At any other point, there would be a tangential component to the net force.

#### blackboy

I don't understand that because I have not learned about force or tension yet.

#### rock.freak667

Homework Helper
I don't understand that because I have not learned about force or tension yet.
From http://en.wikipedia.org/wiki/Tension_(mechanics)" [Broken]]:

tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. It is the opposite of compression

Basically if you put a weight on a string, which will cause it to extend, the tension is a force which resists this extension.

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#### einstein_a_go_go

Also (slightly off the question, but relevant to pendulums), a pendulum only does small oscillations. 45 degrees is a bit large for the oscillations to be that of a pendulum (think how wide grandfather clocks are / aren't!)

#### blackboy

@einstein- I just made that problem up, so you guys could help me.

Is there a way to show this with just kinematics and not dynamics? In the book this chapter is before the dynamics part and I think they expect me to know why with what we have learned.

Sorry for being a not being understanding.

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#### Brasi333

blackboy- I am not a physics pro or homework helper, so take what I say with a grain of salt.

At all times, there are only 2 forces on the pendulum, that of gravity, and that of the tension from the pendulum arm. The tension from the pendulum arm is always directed towards the center of the arc traced by the pendulum. This tension results in centripetal (center-seeking) acceleration. You know when you're riding shotgun in a car and the driver makes a hard left turn, and you feel the pressure of the car door or armrest on your right side? This is a result of the car accelerating towards the center of the arc-turn it is making (this is best conceptualized from a top-down view). This is another form of centripetal acceleration. Centripetal acceleration is always directed towards the center, so by itself it cannot result in tangential acceleration (which, by definition, is acceleration from a force that is tangential to the circle).

The other force involved, gravity, is of course always pointed directly down. Imagine the pendulum at 90 degrees from the vertical position, i.e. it is sticking directly out to the left or right. At this point the gravity force is completely tangent to the instantaneous velocity of the pendulum (which would also be pointed directly down here. In fact, this is a good way to explain tangential acceleration, it must be parallel to the direction of the instantaneous velocity.) As the pendulum falls, its instantaneous velocity changes, from pointing directly down when it first starts falling, to being horizontal at the bottom of the arc. Before it hits the bottom, while its still falling, the combined forces of gravity and tension result in a tangential acceleration (this is the hardest part to visualize, best done by drawing the vectors forces from each and combining them together).

However, at the bottom the force of gravity is completely perpendicular to the motion of the pendulum. Gravity doesn't contribute to purely horizontal motion, so no tangential acceleration either. This is basically what dx said, but it seemed like you needed more of a background, so I went off into a rant.

#### blackboy

Thanks for all replies, but I really want a way to understand it without dynamics. My book says that there is tangent component of g at the top or at the bottom. I don't know what that means.

#### dx

Homework Helper
Gold Member
Acceleration is the result of forces. You can't deduce anything about the acceleration of the pendulum bob without thinking about the forces on it.

#### blackboy

Say it wasn't a pendulum, just an object moving around in a circle, can you answer why the tangential acceleration is 0 at the top or bottom or is it the same thing?

#### dx

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Gold Member
If it's just moving around the circle, what is it's tangential speed at various times? It's speed could be changing, and either you have to know how it is changing, or you have to know the forces so you can calculate how it is changing.

#### blackboy

Looking at this now I forgot to say it was moving in a vertical circle. What does gravity have to do with the 0 tangential acceleration at the top and bottom?

#### dx

Homework Helper
Gold Member
Didn't I explain that already in post #4? What didn't you understand in that?

#### blackboy

I don't know anything about dynamics yet, so I didn't understand.

#### ideasrule

Homework Helper
OK, so here's an intuitive understanding that should require no knowledge of physics.
Gravity always pulls down; that's pretty obvious. When a string is stretched, it pulls back along its length. In a pendulum, that would be towards the pivot.

At the bottom of the pendulum's arc, gravity pulls the bob down while tension pulls it up. There's nothing pulling sideways--"tangentially", as you called it--so there's no reason for the bob to change its speed. That's why tangential acceleration is zero.

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