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#1 2007-05-26 21:16:20

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Object sliding down a straight frictionless slope

I was toying with the following after reading the thread on the brachistochrone problem started by Mikau.

Suppose an object of mass m slides under its own weight down a straight, frictionless slope inclined at angle θ to the vertical. Suppose the object starts from rest at the top of the slope, and let the vertical height of the slope be h.

Let the vertical acceleration of the object by a and the normal reaction of the slope on the object be R. Then ma = mgRsinθ, and the time taken for it to reach the bottom of the slope is √(2ha). During this time, it will have travelled a horizontal distance of htanθ with horizontal acceleration Rcosθ/m.

However I could have derived this by resolving forces on the object perpendicular to the plane of the slope. Duh!

Anyway, this gives

Last edited by JaneFairfax (2010-02-21 22:19:42)

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#2 2007-05-26 23:52:51

LQ
Real Member
Registered: 2006-12-04
Posts: 1,285

Re: Object sliding down a straight frictionless slope

the vector sum must be composed of a horizontal and a vertical vector, their sum is always constant in this case. i think. At the end of the slope that is.

Last edited by LQ (2007-05-27 20:25:11)


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#3 2007-05-27 00:09:46

MathsIsFun
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Registered: 2005-01-21
Posts: 7,711

Re: Object sliding down a straight frictionless slope

You are right to be worried ... if the angle were only 1 degree off horizontal, in order to accelerate downwards at 1g the object must accelerate at 57g horizontally, where does it get that energy from? Test that case.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#4 2007-05-27 00:18:05

Identity
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Registered: 2007-04-18
Posts: 934

Re: Object sliding down a straight frictionless slope

Hmmm, this is probably counter-intuitive because there is no such thing as a frictionless surface in our real world. Perhaps weird stuff just happens on those surfaces... why don't you try incorporating a little friction into the equation and see what happens.

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#5 2007-05-27 00:38:40

luca-deltodesco
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Registered: 2006-05-05
Posts: 1,470

Re: Object sliding down a straight frictionless slope

plane inclined to vertical at angleθ

particle of mass 'm' under gravity 'g'

normal contact force = mgsinθ, acting at angle θ with horizontal

only dealing with vertical components, letting 'a' be vertical acceleration on particle:

let h be the vertical distance it has to travel: let go from rest.

that t, is the time itll take to reach the bottom, with θ = 0, it will take √(2h/g) seconds,
with for example θ = 45°, it will take √(2h/0.5g), in otherwords, twice as lon

Last edited by luca-deltodesco (2007-05-27 00:50:18)


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#6 2007-05-27 00:41:25

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: Object sliding down a straight frictionless slope

The acceleration is along the slope, at a=gCosθ
The total distance is s=h/Cosθ

So the time for travel is

t=√(2s/a)= √[2h/(Cosθ g Cosθ) = Secθ √(2h/g)

The longer the distance, the larger the angle θ, and the larger the secθ, finally the longer the time.


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#7 2007-05-27 00:50:24

luca-deltodesco
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Registered: 2006-05-05
Posts: 1,470

Re: Object sliding down a straight frictionless slope

test my hypothesis of it being at same speed:

if it was vertical, then you would simply have t = √(2h/g), v = √(2hg) ms-¹, v² = 2hg

vertical acceleration:


horizontal acceleration:

time taken:

v = u + at


so the speeds are equal no matter what the slope when it reaches the bottom, only its moving in a different direction due to the normal contact force, which is backed up by simple energy equation

mgh = 0.5mv², since there is no energy lost through friction, it will be the same, and ive shown it mathematically above with the accelerations due to the normal contact force

Last edited by luca-deltodesco (2007-05-27 00:57:23)


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#8 2007-05-27 07:56:10

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Re: Object sliding down a straight frictionless slope

Luca is right.


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