perplexus.info :: Science : Bowl and Rod

Bowl and Rod (Posted on 2004-12-31)

Given a solid bowl (an idealized hemisphere, the opening facing straight up), let a slender rod rest in the bowl supported at two points: a point on the bowl's edge, and some point on the bowl's interior surface.

The bowl's edge exerts a force on the rod which is perpendicular to the rod's length.

The bowl's interior exerts, on the rod's end, a force which is perpendicular to the bowl's surface (at the point where they meet).

If the rod's length is three times the radius of the bowl, what is the angle between the rod, and the plane of the bowl's edge?

Submitted by Bractals

Rating: 2.0000 (2 votes)

Solution: (Hide)

Let E and S represent the forces where the bowl touches the rod (E at edge and S at interior surface). Let W be the weight of the rod, R the radius of the bowl, L = 3R the length of the rod, and x the angle. Then

1) S*cos(2x) = E*sin(x)
------ Forces perpendicular to gravity
2) W*sin(x) = S*cos(x)
------ Forces parallel to rod's length
3) W*[(L/2)*cos(x)] = E*[(2R)*cos(x)]
------ Torques about point of interior contact

Giving 8*cos(x)^2 - 3*cos(x) - 4 = 0 or

cos(x) = [ 3 + sqrt(137) ] / 16 or

x ~= 23.21 degrees.

Note: cos(x) = [ 3 - sqrt(137) ] / 16 does not satisfy the physical problem.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
answer	K Sengupta	2008-02-08 05:56:27
re: FBD	Larry	2005-01-06 22:12:45
re(2): FBD	nikki	2005-01-06 20:59:53
re: FBD	Hugo	2005-01-06 20:56:55
FBD	nikki	2005-01-06 19:19:01
re: Another method, Another answer	Bractals	2005-01-04 22:09:15
Another method, Another answer	Larry	2005-01-03 02:04:29
re(3): Gosh	Larry	2005-01-03 01:05:03
re(2): Gosh	Hugo	2005-01-02 14:55:47
re: Gosh	Larry	2005-01-01 19:56:47
Gosh	CeeAnne	2005-01-01 17:44:10
Solution	Larry	2005-01-01 14:03:19
One more equation	Larry	2005-01-01 12:35:26
re: This might help.	Penny	2005-01-01 10:32:11
This might help.	Bractals	2005-01-01 05:08:43
re: Intuition Says	Larry	2005-01-01 03:07:21
Intuition Says	CeeAnne	2005-01-01 00:47:16
Problem set up	Larry	2004-12-31 17:33:26

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