1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Equilibrium of a stiff plate on inclined planes

  1. May 17, 2018 #1
    1. The problem statement, all variables and given/known data
    A thin stiff uniform rectangular plate with width L (L =AB) is lying on two inclined surface as shown in Fig. 3-1. The angle between the horizontal surface and the left inclined surface is α, and that between the horizontal surface and the right inclined surface is β. It is assumed that the plate length is sufficiently larger than L and both edges of the plate are smooth enough to neglect friction.

    (1) In case of α+β = 90°, the static balance shown in Fig.3-1 is
    (a) possible only provided α≠β
    (b) always possible
    (c) always impossible
    1526215844711.jpg

    2. Relevant equations


    3. The attempt at a solution
    From the picture, at point A and B,
    Calculating in horizontal axis : NA(normal force at a)*sinα = NBsinβ
    vertically direction : NAcosα+NBcosβ=mg
    And I took a moment at point B, so NA*L = mg*Lcosα.
    And I tried to substitue all variables into α and β only, but I got quadratic equation that looked like wrong : (cosα)^2+cosα(sinα)/sinβ-2=0 because I have solved with x=-b+-b^2.... but it couldn't be solved.

    Which way should I do to solve this problem?
     
  2. jcsd
  3. May 17, 2018 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    What angle does NA make to the plate?
    How far along the plate is its mass centre from B?
    What angle does the weight of the plate make to the plate?
     
  4. May 17, 2018 #3
    (1) I think it is α
    (2) Isn't it L/2?
    (3) α ?
     
  5. May 17, 2018 #4

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That would make NA horizontal.
    So why no /2 in your equation?
    I see no reason for it to do so. Start with this question: what angle does the plate make to the horizontal?
     
  6. May 17, 2018 #5
    (1) I think the angle that NA makes to plate is 90 because it's normal force.
    (2) I see.
    (3) I really can't find the angle which the plate make to the horizontal. Could you give me a hint please?
     
  7. May 17, 2018 #6

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The end of the plate rests on the slope. To decide where the normal is you have to find the contact plane, i.e. a flat plane that fits between them without intersecting either.
    It is unknown. The question is whether there is an angle which makes the system stable.
     
  8. May 17, 2018 #7
    (1) Isn't NA perpendicular to the plate? It looks same as a typical rectangular object on a inclined plane problems

    (2) I think I found the angle which plate makes to the horizontal now, it is 90-β.
    But I don't know how to find the answer to your latest questions.
     
  9. May 17, 2018 #8

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    In a typical object on an inclined plane problem, the normal is normal to the plane. If it is a rectangular object flush to the plane then it will be normal to that too, but here it is not.
    See section 1 in https://www.physicsforums.com/insights/frequently-made-errors-mechanics-friction/
    Reread what I wrote in post #6. The slope of the plate is unknown. You can see this by drawing the diagram with the two slopes unchanged but sliding the plate to a different angle.
     
  10. May 17, 2018 #9
    Ah, I see. So the NA makes α angle with vertical direction or 90° to the left inclined plane.

    (2) I understand that you meant slope of both planes, if I understand correctly I think there is a angle which make this system possible. Because there are normal forces to make it become steady.
     
    Last edited: May 17, 2018
  11. May 17, 2018 #10

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Yes.
    That is what the question is asking, and determining it is not trivial.

    Can you answer this: if three forces act on a body in equilibrium, what can you say about the lines of action of those forces?
     
  12. May 17, 2018 #11
    The sum of vectors is zero
    The Vectors cross or intersect at same point and they are in the same plane.
     
  13. May 17, 2018 #12

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That's the one.
    You know α and β add up to π/2 in the first case. Draw yourself a diagram that represents that better, showing the three forces intersecting.
    It will help to create an unknown, θ, for the slope of the plate.
     
  14. May 17, 2018 #13
    The angle which the plate makes to horizontal, θ, is 90°-β. How could this help to solve problems?
     
  15. May 17, 2018 #14

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    For the third time, at least, θ is not known.
    Draw the two slopes at some angles α and β. Mark a point A for one end of the plate on the left hand slope. You can now put B anywhere you like on the right hand slope, so θ is any value from -α (i.e. sloping down to the right, as shown) to +β.

    Your task is to say whether, for the given α and β, there is some θ for which the arrangement is stable.
     
  16. May 17, 2018 #15
    there is
     
  17. May 17, 2018 #16

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You are only guessing. You have not shown that.
    I should clarify for you that this is a question about stability, which is not the same as equilibrium.
    A pencil on its point could, in principle, be in equilibrium, but it cannot be stable. The slightest perturbation will make it fall over.

    Draw the diagram I described in post #12.
    Call the point at the base where the slopes meet O and centre of the plate G. Show the three intersecting forces, with the normals at right angles to the slopes and α+β=π/2. You should then be able to answer this key question: what is the distance OG?
     
    Last edited: May 17, 2018
  18. May 17, 2018 #17
    I have drawn my diagram, it looks like rectangle which has L as diagonal. I think OG is L/2.
     
  19. May 18, 2018 #18

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Right!
    What is the orientation of the other diagonal through the mass centre?
    If the plate were to slip a little, with one end going down its slope and the other end up its slope, what path would the mass centre move along? What would happen to the height of the mass centre?
     
  20. May 18, 2018 #19
    The other diagonal passes through the mass centre of plate.

    I have drawn the left end plate going down and the other going up. The mass centre of the new diagram is located at same position of the first diagram. Right?
     
  21. May 18, 2018 #20

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Sure, but I asked what its orientation is, e.g. what angle does it make to the vertical?
    My question was what path would the mass centre take if the plate were to slip. For that, remember what you found in post #17.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted