Problems and Solutions Section 1.1 (1.1 through 1.26)
1.1 Consider a simple pendulum (see Example 1.1.1) and compute the magnitude of the
restoring force if the mass of the pendulum is 3 kg and the length of the pendulum is 0.8 m.
Assume the pendulum is at the surface of the earth at sea level.
Solution: From example 1.1.1, the restoring force of the pendulum is , which has
maximum value
1.2 Compute the period of oscillation of a pendulum of length 1.2 m at the North Pole where the
acceleration due to gravity is measured to be 9.832 m/s2.
Solution: The natural frequency and period can be computed with the following
relationships:
1.3 The spring of Figure 1.2, repeated here as Figure P1.3, is loaded with mass of 10 kg and
the corresponding (static) displacement is 0.012 m. Calculate the spring's stiffness.
Solution:
From the free-body diagram and static
Free-body diagram:
equilibrium:
)
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,1.4 The spring of Figure P1.3 is successively loaded with mass and the corresponding (static)
displacement is recorded below. Plot the data and calculate the spring's stiffness. Note
that the data contain some error. Also calculate the standard deviation.
m(kg) 10 11 12 13 14 15 16
x(m) 1.14 1.25 1.37 1.48 1.59 1.71 1.82
Solution:
Free-body diagram: From the free-body diagram and static
equilibrium:
The sample standard deviation in
computed stiffness is:
Plot of mass in kg versus displacement in m
Computation of slope from mg/x
m(kg) x(m) k(N/m)
10 1.14 86.05
11 1.25 86.33
12 1.37 85.93
13 1.48 86.17
14 1.59 86.38
15 1.71 86.05
16 1.82 86.24
1.5 Consider the pendulum of Example 1.1.1 and compute the amplitude of the
restoring force if the mass of the pendulum is 2 kg and the length of the pendulum is
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, 0.5 m if the pendulum is at the surface of the moon.
Solution: From example 1.1.1, the restoring force of the pendulum is ,
which has maximum value
1.5
1.6 Consider the pendulum of Example 1.1.1 and compute the angular natural frequency
(radians per second) of vibration for the linearized system if the mass of the
pendulum is 3 kg and the length of the pendulum is 0.8 m if the pendulum is at the
surface of the earth. What is the period of oscillation in seconds?
Solution: The natural frequency and period are:
1.7 Derive the solution of and plot the result for at least two periods for the case with
ω n = 2 rad/s, x 0 = 1 mm, and v 0 = mm/s.
Solution:
Given:
(1)
Assume: . Then: and . Substitute into equation (1) to get:
Thus there are two solutions:
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, The sum of x 1 and x 2 is also a solution so that the total solution is:
Substitute initial conditions: x 0 = 1 mm, v 0 = mm/s
Therefore the solution is:
Using Mathcad the plot is:
Copyright © 2015 Pearson Education Ltd.
1.1 Consider a simple pendulum (see Example 1.1.1) and compute the magnitude of the
restoring force if the mass of the pendulum is 3 kg and the length of the pendulum is 0.8 m.
Assume the pendulum is at the surface of the earth at sea level.
Solution: From example 1.1.1, the restoring force of the pendulum is , which has
maximum value
1.2 Compute the period of oscillation of a pendulum of length 1.2 m at the North Pole where the
acceleration due to gravity is measured to be 9.832 m/s2.
Solution: The natural frequency and period can be computed with the following
relationships:
1.3 The spring of Figure 1.2, repeated here as Figure P1.3, is loaded with mass of 10 kg and
the corresponding (static) displacement is 0.012 m. Calculate the spring's stiffness.
Solution:
From the free-body diagram and static
Free-body diagram:
equilibrium:
)
Copyright © 2015 Pearson Education Ltd.
,1.4 The spring of Figure P1.3 is successively loaded with mass and the corresponding (static)
displacement is recorded below. Plot the data and calculate the spring's stiffness. Note
that the data contain some error. Also calculate the standard deviation.
m(kg) 10 11 12 13 14 15 16
x(m) 1.14 1.25 1.37 1.48 1.59 1.71 1.82
Solution:
Free-body diagram: From the free-body diagram and static
equilibrium:
The sample standard deviation in
computed stiffness is:
Plot of mass in kg versus displacement in m
Computation of slope from mg/x
m(kg) x(m) k(N/m)
10 1.14 86.05
11 1.25 86.33
12 1.37 85.93
13 1.48 86.17
14 1.59 86.38
15 1.71 86.05
16 1.82 86.24
1.5 Consider the pendulum of Example 1.1.1 and compute the amplitude of the
restoring force if the mass of the pendulum is 2 kg and the length of the pendulum is
Copyright © 2015 Pearson Education Ltd.
, 0.5 m if the pendulum is at the surface of the moon.
Solution: From example 1.1.1, the restoring force of the pendulum is ,
which has maximum value
1.5
1.6 Consider the pendulum of Example 1.1.1 and compute the angular natural frequency
(radians per second) of vibration for the linearized system if the mass of the
pendulum is 3 kg and the length of the pendulum is 0.8 m if the pendulum is at the
surface of the earth. What is the period of oscillation in seconds?
Solution: The natural frequency and period are:
1.7 Derive the solution of and plot the result for at least two periods for the case with
ω n = 2 rad/s, x 0 = 1 mm, and v 0 = mm/s.
Solution:
Given:
(1)
Assume: . Then: and . Substitute into equation (1) to get:
Thus there are two solutions:
Copyright © 2015 Pearson Education Ltd.
, The sum of x 1 and x 2 is also a solution so that the total solution is:
Substitute initial conditions: x 0 = 1 mm, v 0 = mm/s
Therefore the solution is:
Using Mathcad the plot is:
Copyright © 2015 Pearson Education Ltd.
