Mechanical Engineering Principles

178 MECHANICAL ENGINEERING PRINCIPLES

pd

D 1

D 2

p

q

Figure 15.15

as shown in Figure 15.15, and

d=

(D 1 +D 2 )

2

Ifμis the coefficient of friction up the slope, then
let tanλ=μ.
Referring now to Figure 15.16, the screw jack can
be analysed.

P

N

W

F q

q

Motion

Figure 15.16

Resolving normal to the plane gives:

N=Wcosθ+Psinθ( 15. 24 )

Resolving parallel to the plane gives:

Pcosθ=F+Wsinθ( 15. 25 )

and F=μN ( 15. 26 )

Substituting equation (15.26) into equation (15.25)
gives:

Pcosθ=μN+Wsinθ( 15. 27 )

Substituting equation (15.24) into equation (15.27)

gives:

Pcosθ=μ(Wcosθ+Psinθ)+Wsinθ

Dividing each term by cosθand remembering that

sinθ

cosθ

=tanθgives:

P=μ(W+Ptanθ)+Wtanθ

Rearranging gives:

P( 1 −μtanθ)=W(μ+tanθ)

from which, P=

W(μ+tanθ)

( 1 −μtanθ)

=

W(tanλ+tanθ)

( 1 −tanλtanθ)

sinceμ=tanλ

However, from compound angle formulae,

tan(λ+θ)=

(tanλ+tanθ)

( 1 −tanλtanθ)

Hence, P=Wtan(θ+λ) ( 15. 28 )

However, from Figure 15.15,

tanθ=

p

πd

hence P=

W

(

μ+

p

πd

)

(

1 −

μp

πd

) ( 15. 29 )

Multiplying top and bottom of equation (15.29) by

πdgives:

P=

W(μπd+p)

(πd−μp)

( 15. 30 )

Theuseful work donein lifting the weightW a

distance ofp

=Wp ( 15. 31 )

From Figure 15.15,the actual work done

=P×πd

=

W(μπd+p)

(πd−μp)

×πd ( 15. 32 )