# A piston is connected by a rod of #14 cm# to a crankshaft at a point #5 cm# away from the axis of rotation. Determine how fast the crankshaft is rotating when the piston is 11 cm away from the axis of rotation and is moving toward it at 1200 cm/s?

##### 1 Answer

#### Explanation:

This problem is asking us to find

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Let

So we know that:

#(dQ)/dt]_(Q= 11) = -1200"cm"/s#

Before we start, let's write out a formula for theta given what we know, using the Law of Cosines:

#14^2 = 5^2 + Q^2 - 2(5)(Q)cos(theta)#

#10Qcostheta = Q^2 - 171#

#10costheta = Q - 171/Q#

Now, take the derivative of this with respect to time:

#-10sintheta * (d theta)/dt = (dQ)/dt + 171/Q^2 * (dQ)/dt#

Since we already have values for

To do that, let's use the Law of Cosines again, but manipulate it further until we get

#14^2 = 5^2 + 11^2 - 2(5)(11)cos(theta)#

#50 = -110cos(theta)#

#-5/11 = cos(theta) = sqrt(1-sin^2theta)#

#25/121 = 1-sin^2theta#

#96/121 = sin^2theta#

#(4sqrt6)/11 = sintheta#

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Now we know that when

This is all we need to solve for

#-10sintheta * (d theta)/dt = (dQ)/dt + 171/Q^2 * (dQ)/dt#

#-10((4sqrt6)/11)*(d theta)/dt = -1200 + 171/121(-1200)#

#-(40sqrt6)/11*(d theta)/dt = -1200(292/121)#

#(d theta)/dt = (1200 * 292 * 11)/(40sqrt6 * 121)" ""rad"/s#

#(d theta)/dt = (8760)/(11sqrt6) = (8760sqrt6)/66 = (1460sqrt6)/11 " ""rad"/s#

Or if you prefer decimal form:

#(d theta)/dt = 325.114 " " "rad"/s#