Answer:
a) k = 40 N/m
b) See attachment
c) Amp y = 0.1 m , T_a = 0.67 s
d) T_theo = 0.62831
e) RE = 6.63 %
Explanation:
Given:
- mass of the spring m = 0.4 kg
- Displacement - time values
- Force-displacement graph
Find:
a. Determine the spring constant from the graph.
b. Make a position vs. time graph for the spring’s motion.
c. Determine the amplitude and period of the spring’s motion from your graph.
d. Calculate the period of the spring’s motion using the spring’s period equation.
e. Determine the percent error by comparing the period from the graph (experimental value) to the calculated value (accepted value).
Solution:
a)
- Use the Force-displacement graph and pick two points on the line A and B:
A = ( 0.006 , 1.78 ) & B = ( 0.083, 4.75 )
- We know from Hook's Law that:
y = k*x
k = y/x ------ gradient of the plot will give us the constant k
- So,
k = (y_2 - y_1) / (x_2 - x_1)=(4.75-1.78)/(0.083-0.006)=40 N/m
b) See attachment
c) Using the plot of b:
The maximum amplitude of displacement y = 0.1 m
The time period of the displacement T_a = 2 s / 3 cycles = 0.67s
d)
- The formula for computing time period is:
T_theo = 2*pi*sqrt(m/k)
T_theo = 2*pi*sqrt(0.4/40) = 0.62831 s
e)
- Comparing values by computing the relative error:
RE = ( T_a - T_theo) / T_theo * 100
RE = (0.67 - 0.62831) / 0.62831 * 100 = 6.63 %
- The value obtained is within the 10% tolerance of experimental Error. Hence, we can conclude that the experiment was conducted reasonably.
