An Introduction to Physical Science
14th Edition
ISBN: 9781305079137
Author: James Shipman, Jerry D. Wilson, Charles A. Higgins, Omar Torres
Publisher: Cengage Learning
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Chapter 4, Problem 15SA
A simple pendulum as shown in ● Fig. 4.24 oscillates back and forth. Use the letter designations in the figure to identify the pendulum’s position(s) for the following conditions. (There may be more than one answer. Consider the pendulum to be ideal with no energy losses.)
- (a) Position(s) of instantaneous rest ___
- (b) Position(s) of maximum velocity ___
- (c) Position(s) of maximum Ek ___
- (d) Position(s) of maximum Ep ___
- (e) Position(s) of minimum Ek ___
- (f) Position(s) of minimum Ep ___
- (g) Position(s) after which Ek increases ___
- (h) Position(s) after which Ep increases ___
- (i) Position(s) after which Ek decreases ___
- (j) Position(s) after which Ep decreases ___
Figure 4.24 The Simple Pendulum and Energy
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We will now determine how the 1/3 rule comes about.
Consider a spring of mass ms which is attached to a wall and oscillates on a frictionless surface as shown below. The spring’s mass is uniformly distributed along the length of the spring.
We will start with the infinitesimal form of kinetic energy, i.e. dKE = ½ (dms )v2. This formula will apply to an infinitesimal segment of the spring of length dx and mass dms as indicated below.
For any point on the spring, the velocity of oscillation will be given by v = (ve/L)x where ve is the velocity of the spring at its end where the mass m is attached, and L is the stretched length of the spring at that instant. Thus, when x = 0 then v = 0, and when x = L/2 then v = ½ ve.
Hint: Figure out how to relate dms to dx and then integrate both sides of the infinitesimal kinetic energy equation to get an equation for the kinetic energy of the spring that includes ms/3.
Problem 10: The restoring force in a Hooke's Law spring is a conservative force, and hence, the work done by that force is represented by a potential-
energy function. Consider a spring with spring constant
k. One end of the spring is fixed at the origin of the coordinate system, and the other is attached to an oscillating block with mass
m. The diagram below shows the potential energy function for the spring and the total mechanical energy of the system.
14.0
12.0
10.0
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8.0
6.0
4.0+
2.0
0.0
10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0
x(cm)
Part (a) What is the equilibrium position, in centimeters, of the block?
Xeq=18
✓ Correct!
Part (b) What is the position, in centimeters, of one of the turning points of the spring-mass system.
Xturn = 14 ✔ Correct!
Part (c) What is the value, in newtons per meter, of the spring constant?
k=
N/m
TL
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sin()
cos()
tan()
cotan() asin() acos() E 1
4
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acotan()
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2
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atan()
cosh()
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.
tanh()
cotanh()
ODegrees…
The diagram shows a horizontal spring attached to a block. The block’s equilibrium position is at C, and the block oscillates between the points A and E with no friction from the floor. As the block moves from C to E, which of the following statements is true? Usp is the spring potential energy of the spring and K is the kinetic energy of the block.
Chapter 4 Solutions
An Introduction to Physical Science
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