How does the acceleration of the pendulum change as it swings from its highest point, through its lowest point, and, again, back up to its highest point?
Acceleration of a Simple Pendulum
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http://upload.wikimedia.org/wikipedia/commons/2/21/Pendulum_animation.gif
yaay wiki.
since it's a pendulum and you always have Tension as a force, accel is never 0. at the lowest point, all the acceleration is radial. velocity is max, no net forces in the tangential direction, no acceleration in the tangential direction. at the peaks resolve your pendulum weight mg into tangential and radial directions. the radial one =tension, the tangential one is your net force, =ma.
methinks.
yaay wiki.
since it's a pendulum and you always have Tension as a force, accel is never 0. at the lowest point, all the acceleration is radial. velocity is max, no net forces in the tangential direction, no acceleration in the tangential direction. at the peaks resolve your pendulum weight mg into tangential and radial directions. the radial one =tension, the tangential one is your net force, =ma.
methinks.
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As the pendulum falls, it accelerates. It reaches its maximum velocity at the very bottom of the path. As it begins to move back up, it begins decelerating. Repeat. Pretty sure this is how it goes. Seems intuitive.
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I was going to post the same gif!
how does the value of tension change as it swings from its highest amplitude to lowest amplitude?
I am drawing out diagrams, and I can't wrap my head around it.
I do know, though, at the lowest point, T=mg.
I am drawing out diagrams, and I can't wrap my head around it.
I do know, though, at the lowest point, T=mg.
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Think about swinging on a swing set. When you are at the top of your arc and the seat stops for a brief moment, the chain has a lot of slack (not very much tension). When you are at the bottom of your arc and moving fastest, the chain is taught (high tension).
This is explained by math when you consider that tension is a reactionary force that opposes part of the weight (the vector component that is colinear with tension). Tension offsets the cosine component of the weight, so as the angle increases (you rise to a greater height), the cosine component of the weight decreases and the amount of tension necessary to offset it (react to it) goes down.
Tension changes direction and magnitude during the cycle of the pendulum.
Are you sure about the acceleration part? I'm struggling with the simple pendulum magnitude of acceleration. At the bottom point of the swing, when you have maxKA and minPE and max v (that part I get), but minimum magnitude of acceleration, and at the top of the swing, you have max acceleration?
Can someone help explain that? (I'm still not getting it after reading the answer from one of the TBR CBTs)
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At the top of the swing you have max acceleration due to gravity, at the bottom the least.
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I was taught that gravity was constant and it was the tension that changed, causing the net force to change. The net force is F = mg sin(theta), because the tension is cancelled out by mg cos(theta). As you get closer to the bottom of your path, theta gets closer to 0, so the tension is cancelling out more of the gravitational force. Once at the bottom the two are opposing one another and cancel.
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To build on that, if you were able to take your pendulum all the way up so that it was 90 degrees from its original position, assuming uniform distribution of mass (like a sphere) sin theta would again equal 1. So, mg is once again acting "straight down" on your pendulum's center of gravity. At that point, as long as no other force acts to push it up, it has "no choice" but to accelerate downwards. This is partially because, if tension and mg are exactly perpendicular to each other, with tension being exactly on the x-axis and mg being exactly on the y-axis, the "sideways" tension can't influence the "straight down" mg. So, momentarily independent of tension, it accelerates down at max acceleration, with tension playing an increasing role in the y-component of the force vector. This is why in the real world, you "bounce" for a sec when you're on a swing and reach the maximum displacement, or the top of your arc.
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that....is beautiful. tear.
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Gravity is constant. Force of gravity tangential to the pendulum isn't: mgsin(theta). At the bottom there is no tangential force of gravity but there is a force of tension equal to the speed attained by the pendulum squared divided by the length of the pendulum times the mass added to the the weight of the pendulum.
L-Lcos(theta) = h
mg(L-Lcos(theta)) = .5 m x v^2
2mg(L-Lcos(theta)) = m x v^2
2g(1-cos(theta)) = v^2 / L
cos(theta) < 2(1-cos(theta)) + 1
Tension at top < Tension at the bottom
Except for Theta = 0 where Force of tension at angle theta = force of tension at the bottom.
L-Lcos(theta) = h
mg(L-Lcos(theta)) = .5 m x v^2
2mg(L-Lcos(theta)) = m x v^2
2g(1-cos(theta)) = v^2 / L
cos(theta) < 2(1-cos(theta)) + 1
Tension at top < Tension at the bottom
Except for Theta = 0 where Force of tension at angle theta = force of tension at the bottom.
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