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Question

A horizontal force F is applied to a homogeneous rectangular block of mass m, width b and height H. The block moves with constant velocity; the coefficient of friction is μk
(a) What is the greater height h at which the force F can be applied so that the block will slide without tipping over?
(b) Through which point on the bottom face of the block will the resultant of the friction and normal forces act if h=H/2?
(c) If the block is at rest and coefficient of static friction is μk what are the various criteria for which sliding or tipping occurs?

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Solution

In the absence of any external force in horizontal direction the normal reaction N passes through the centre of mass of the block; when force F is applied, normal reaction shifts in the direction of applied force F.
Since right part of body is having tendency to lift from surface, at the instant of tipping over about the edge the normal reaction passes through edge. From the conditions of equilibrium
In horizontal direction f=F=μkmg
In vertical direction N=mg
Balancing torque about edge Fhmgb2
or h=mgb2F=mgb2μkmg=b2μk
We can solve the problems by another approach also as the resultant of friction force (μkN) and the normal reaction must pass through the same point through which F passes, since three coplanar forces keeping a body in equilibrium pass through a common point i.e they should be concurrent.
From Lami's theorem:
Fsin90o=Fsin(180o+θ)=mgsinsin(90o+θ)
Fsinθ=mgcosθortanθ=Fmg=μkmgmg=μk
From figure it is clear tanθ=μkNN=b/2h or b/2h=μkh=b2μk
(b) In this case we cannot take normal reaction at the edge
Let normal reaction acts at a distance x from the line of action of mg
Torque of all the forces about C must be zero
(μkN×H2)=Nx
x=μkH2
Alternatively we can proceed as in part (a). Resultant of μk,N,N passes through C the interaction of F and mg
Therefore from geometry of the figure.
xH/2=tanθ=μk;x=μkH2
(c) As the point of appication of force is raised higher, the location of the line of action of the normal reaction N moves to the left. In the limiting case (when the block is about to trip over), x=b/2. The normal reaction passes through edge as shown in figure.
Initial toppling torque about edge:
Fhmax=mgb2orhmax=mgb2F...(i)
For initial sliding, in horizontal direction: F=μN
In vertical direction: N=mg
Hence, F=μmg
If sliding and tipping are equally likely to occur, we can eliminate F eqs. (i) and (ii) to get hmax=b2μ
which is independent of weight mg, height H of the body and applied force F.

1028889_981990_ans_905b197b8c8f465cafe20463b22d8aa3.PNG

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