Chapter 7 The Conservation of Energy

124
Chapter 7
Picture the Problem Let the system
consist of Earth and the child. Then
Wext = 0. Choose Ug = 0 at the child’s
lowest point as shown in the diagram to
the right. Then the child’s initial energy
is entirely kinetic and its energy when
it is at its highest point is entirely
gravitational potential. We can
determine h from conservation of
mechanical energy and then use
trigonometry to determineθ.
Using the diagram, relate θ to h and
L:
Apply conservation of mechanical
energy to the system to obtain:
Substituting for Ki and Ug,f yields:
Substitute for h in equation (1) to
obtain:
Substitute numerical values and
evaluate θ :
g = U
0
θ
L L − h
vi r
−1⎛
L − h ⎞ −1⎛
h ⎞
θ = cos ⎜ ⎟ = cos ⎜1−
⎟ (1)
⎝ L ⎠ ⎝ L ⎠
W
= ΔK
+ ΔU
= 0
ext
or, because Kf = Ug,i = 0,
− K + U
i
g, f
= 0
2
1 2
vi
− 2 mvi + mgh = 0⇒
h =
2g
θ = cos
θ =
=
−1
⎟ 2 ⎛ v ⎞ i
⎜
⎜1−
⎝ 2gL
⎠
⎛ −1⎜
⎜
1−
⎝ 2
26°
h
2 ( 3.4m/s)
( 9.81m/s
)( 6.0m)
cos 2
45 •• A ball at the end of a string moves in a vertical circle with constant
mechanical energy E. What is the difference between the tension at the bottom of
the circle and the tension at the top?
⎞
⎟
⎠