Concept explainers
(a)
To plot:The position x of a body oscillating on a spring as a function of time.
(a)
Explanation of Solution
Given:
The equation of position x as a function of time
The values of the constants
The time interval
Calculation:
Using the given values of the variables in the given equation,
On a spreadsheet calculate the values of the position with respect to time and plot a graph as shown.
t in s | x in cm |
0 | 0 |
1 | 0.87054 |
2 | 1.71449 |
3 | 2.50607 |
4 | 3.22109 |
5 | 3.83772 |
6 | 4.33712 |
7 | 4.70403 |
8 | 4.92725 |
9 | 4.99996 |
10 | 4.91993 |
11 | 4.68961 |
12 | 4.31605 |
13 | 3.81064 |
14 | 3.18882 |
15 | 2.4696 |
16 | 1.67494 |
17 | 0.82912 |
18 | -0.042 |
19 | -0.9119 |
20 | -1.7539 |
21 | -2.5424 |
22 | -3.2531 |
23 | -3.8645 |
24 | -4.3579 |
25 | -4.7181 |
26 | -4.9342 |
27 | -4.9996 |
28 | -4.9123 |
29 | -4.6749 |
30 | -4.2947 |
31 | -3.7833 |
32 | -3.1563 |
33 | -2.433 |
34 | -1.6353 |
35 | -0.7876 |
36 | 0.08407 |
Figure 1
Conclusion:
Thus, the position x of the object which undergoes oscillation following the equation
(b)
To measure:The slope of the
(b)
Answer to Problem 114P
The velocity of the object at time
Explanation of Solution
Given:
The
Calculation:
Draw a tangent to the curve at time
Figure 2
From Figure 2, the slope of the tangent (drawn in red) is given by,
Hence the velocity of the object at time
Conclusion:
Thus, the velocity of the object at time
(c)
To calculate:The average velocity for a series of intervals starting from
(c)
Answer to Problem 114P
The average velocities for the time intervals starting at
Explanation of Solution
Given:
The equation for the position of the oscillating particle
The times at which the average velocity is determined
Formula used:
The average velocity of a particle is the rate of change of position of the object during the time interval.
Calculation:
Determine the value of the position of the object
Determine the position of the particle at time
Determine the average velocity for the time interval
Find the position of the particle at time
Determine the average velocity for the time interval
Find the position of the particle at time
Determine the average velocity for the time interval
Find the position of the particle at time
Determine the average velocity for the time interval
Find the position of the particle at time
Determine the average velocity for the time interval
Find the position of the particle at time
Determine the average velocity for the time interval
Conclusion:
Thus, the average velocities for the time intervals starting at
(d)
To compute:
(d)
Answer to Problem 114P
The value of
Explanation of Solution
Given:
The equation for the position of the oscillating particle
Formula used:
The velocity of a particle is the first derivative of the position with respect to time and is given by,
Calculation:
Differentiate the given equation with respect to time.
Substitute
Conclusion:
The value of
(e)
To compare: the results of parts (c) and (d) and explain why the part(c) results approach part (d) result.
(e)
Explanation of Solution
Given:
Results of part (c)
The average velocities of the particle for the time intervals starting at
are as follows:
Time interval(s) | Average velocity (cm/s) |
0-6.0 | 0.72 |
0-3.0 | 0.86 |
0-2.0 | 0.86 |
0-1.0 | 0.87 |
0-0.50 | 0.87 |
0.25 | 0.87 |
Results of part (d)
The instantaneous velocity of the particle at time
Introduction:
Average velocity is defined as the ratio of change in position to the time interval.
The instantaneous velocity is given by,
As the measured time interval becomes smaller, the average velocity approaches the instantaneous velocity. For a large time interval such as
Conclusion:
Thus, it can be seen that as th magnitude of the measured time intervals decrease, the values of the average velocities approach the value of instantaneous velocity.
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