p08_030.pdf

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Chapter 8 - 8.30
30. The connection between angle θ (measured from vertical – see Fig. 8-29) and height h (measured from the
lowest point, which is our choice of reference position in computing the gravitational potential energy)
is given by h = L (1
−
cos θ )where L is the length of the pendulum.
(a) We use energy conservation in the form of Eq. 8-17.
K 1 + U 1 = K 2 + U 2
cos θ 1 )= 1
0+ mgL (1
−
2 mv 2 + mgL (1
−
cos θ 2 )
This leads to
v 2 = 2 gL (cos θ 2 −
cos θ 1 )=1 . 4m / s
since L =1 . 4m, θ 1 =30 ◦ ,and θ 2 =20 ◦ .
(b) The maximum speed v 3 is at the lowest point. Our formula for h gives h 3 =0when θ 3 =0 ◦ ,as
expected.
K 1 + U 1 = K 3 + U 3
cos θ 1 )= 1
0+ mgL (1
−
2 mv 3 +0
This yields v 3 =1 . 9m/s.
(c) We look for an angle θ 4 such that the speed there is v 4 = v 3 / 3. To be as accurate as possible,
we proceed algebraically (substituting v 3 =2 gL (1
−
cos θ 1 ) at the appropriate place) and plug
numbers in at the end. Energy conservation leads to
K 1 + U 1 = K 4 + U 4
0+ mgL (1
−
cos θ 1 )= 1
2 mv 4 + mgL (1
−
cos θ 4 )
mgL (1
−
cos θ 1 )= 1
2 m v 3
9 + mgL (1
−
cos θ 4 )
−
gL cos θ 1 =
1
2
2 gL (1
cos θ 1 )
9 −
gL cos θ 4
where in the last stepwe have subtracted out mgL and then divided by m . Thus, we obtain
θ 4 =cos − 1 1
9 + 8
9 cos θ 1 =28 . 2 ◦
where we have quoted the answer to three significant figures since the problem gives θ 1 to three
figures.
−
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