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Chapter 8: Q. 54 (page 536)

Rod OArotates about the fixed point Oso that point Atravels on a circle of radius r. Connected to point Ais another rod ABof length L>2r, and point Bis connected to a piston. See the figure. Show that the distance xbetween point Oand point Bis given by

x=rcosθ+r2cos2θ+L2-r2

where θis the angle of rotation of rodOA.

Short Answer

Expert verified

The solution isx=rcosθ+r2cos2θ+L2-r2.

Step by step solution

01

Step 1. Given information 

02

Step 2. Calculation 

Using the Law of Cosines -

L2=r2+x2-2rxcosθ

⇒x2-2rxcosθ=L2-r2

localid="1647191796730" ⇒x2-2rxcosθ+r2cos2θ=L2-r2+r2cos2θ

localid="1647191812441" ⇒x-rcosθ2=r2cos2θ+L2-r2

localid="1647191823942" ⇒x-rcosθ2=r2cos2θ+L2-r2

localid="1647191834254" ⇒x-rcosθ=r2cos2θ+L2-r2

localid="1647191844153" ⇒x=rcosθ+r2cos2θ+L2-r2

Therefore the distance xbetween point Oand point Bis given by -

x=rcosθ+r2cos2θ+L2-r2

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