Find out the smallest force, Mechanical Engineering

Find out the smallest force:

A block weighing 800 N is raised up with the help of two 6o wedges B and C of negligible weights as illustrated in Figure. If the coefficient of static friction is 0.25 for all surfaces of contact, find out the smallest force P to be applied to raise the block A.

810_Find out the smallest force.png

Solution

Let us primary draw the free body diagrams for block A and wedges B and C. Block A is in contact with the vertical wall and the horizontal surface of block B. Thus, the normal reactions N1 and N2 will be perpendicular to the respective surfaces of contact as illustrated in Figure. The direction of movement of block A w.r.t. the wall being vertically upwards, the direction of frictional force shall be vertically downwards as friction opposes the motion. Likewise, the motion of block A w.r.t. wedge B being towards right, the direction of frictional force at the surface of contact of block A and the wedge shall be towards left as illustrated in the free body diagram of block A.

Figure 3.17 : Free Body Diagram of Block A

In the restricting condition, we know F1 = μ N1 and F2 = μ N2. Thus, there are only two unknowns : N1 and N2 and two equations of equilibrium being available, we may find the values of N1 and N2.

∑ Fx  = 0         ∴ N1  - F2  = 0

∴ N1  = F2  = 0.25 N 2

 ∑ Fy  = 0        ∴ N2  - W - F1 = 0

∴ N - 800 - 0.25 N1  = 0

∴ N 2  - 800 - 0.25 (0.25 N 2 ) = 0

∴ N 2  (1 - 0.0625) = 800

∴ N 2    =          800 / 0.9375

          = 853.33 N

Now let us draw the free body diagram of wedge B. Keeping in mind that the frictional forces oppose the motion, and the normal reactions, as the name recommend, are perpendicular to the surfaces of contact; the several forces acting on wedge B shall be as indicated in the free body diagram of wedge B illustrated in Figure.

Figure 3.18 : Free Body Diagram of Wedge B

Applying the equations of equilibrium, we have following:

∑ Fy  = 0 ∴ N3 cos 6o  - N2  - F3 sin 6o  = 0

 ∴ N3  cos 6o  - N 2  - 0.25 N3  sin 6o  = 0 (as F3 = 0.25 N3)

∴ 0.9945 N3  - 853.33 - 0.0261 N3  = 0

∴ N3 = 853.33 /0.9684 = 881.18 N

And, ∑ Fx  = 0

∴ F2  + F3  cos 6o  + N3  sin 6o - P = 0

 ∴ P = F2  + F3  cos 6o + N3  sin 6o

 = 0.25 N 2 + 0.25 N 3  cos 6o + N3  sin 6o

Putting the values of N2 and N3, we have

P = 0.25 × 853.33 + 0.25 × 881.18 cos 6o + 881.18 sin 6o

= 213.33 + 219.09 + 92.11

= 524.53 N

Thus, a force of 524.53 N shall be required to raise the block A.

You may also solve the problem by constructing the triangle of forces R1, R2 and W for block A, where R1 is the resultant reaction of N1 and F1 and R2 is the resultant reaction of N2 and F2.

As the three forces keep the block in equilibrium, the forces should be concurrent. This may be solved graphically or by using Lami's theorem for three concurrent forces.

By applying Lami's Theorem, we obtain:

W / sin (90o + 2 φ)       =         R1 / sin (180o - φ)  =   R2 / sin (90o  - φ)

where

φ = angle of friction

= tan -1  μ = tan -1  (0.25)

= 14.036 o

 ∴ R2  = W cos φ / cos 2 φ

= W cos 14.036o / cos 28.072o

 = (800 × 0.9701 )/0.8823

   = 879.61 N

810_Find out the smallest force.png

                                   Figure

Likewise, for wedge B, there are three forces acting : R2, R3 and P where,

R2 = Resultant reaction of N2 and F2

R3 = Resultant reaction of N3 and F3

The three forces acting shall be as illustrated in Figure

Figure 3.20

By applying Lami's theorem, we have

R2 / sin (90o + φ + 6o )   =        Psin (180o  - 2 φ - 6o ) =         R3 / sin (90o  + φ)

∴ P = R2 sin (2 φ + 6o)/ cos (φ + 6o)

= 879.61 sin 34.072o /cos 20.036o

= 524.53 N

Posted Date: 1/28/2013 5:31:49 AM | Location : United States







Related Discussions:- Find out the smallest force, Assignment Help, Ask Question on Find out the smallest force, Get Answer, Expert's Help, Find out the smallest force Discussions

Write discussion on Find out the smallest force
Your posts are moderated
Related Questions
For an automotive component of your choice that is traditionally metallic, review how it could be re-engineered as a composite? How factors such as fibre & matrix type, fibre volum

Stress Strai n Curves (Tension): When a bar or specimen is subjected to a gradually increasing axial tensile load, the strains and stresses can be found out for number of loa

A semi elliptical spring has 10 leaves in all, with the two full length leaves extending 625 mm. It is 62.5 mm wide and is prepared by strips 6 mm thick. Prepare a design of helica

Illustrate following key terms with suitable example :- (a) Co-planar parallel force. (b) Co-planar non concurrent force. (c) Non co-planar concurrent force.

Q. What do you mean by extrusion? • Extrusion is a process used to create objects of a fixed cross-sectional profile. A material is pushed or drawn through a die of the desir

Method of resolution to find out the resultant force: Some time there is doubt for finding the angle of resultant (θ), the value of the angle θ will be very depending upon the

Example of Equilibrium: Express in terms of θ, β and W the force T necessary to hold weight in equilibrium as shown in the figure given below. Also derive an expression f

Q. Use of high temperature materials in Furnaces? Often, furnace tubing and refractory protected membrane walls are fabricated from plain carbon steel with a minimum of 0.25% c

(a) What are fluids? How are they classified? (b) Classify the several kinds of manometers. (c) Define the following (d) Define metacenter (1) Steady flow (2) Unsteady

how does catalytic ignition operate