In a labratory experiment, a student measured the period of a swing for a 1 -kg pendulum for varing lengths, as shown below.
Length (m) Period (s)
0.05 0.30
0.20 0.90
0.40 1.30
0.60 1.60
0.80 1.80
1.00 2.00
a.) Deterine the period of a pendulm whose length is 0.25 m.
b.) Now the bob is removed from the pendulum and placed on a spring to form a mass-spring system. What would the spring constant of the spring have to be in order to have the same period as that of the pendulum in part a?
a) Plot a graph of Period (P) vs Length (L) and interpolate. If you plot P vs sqrt L, You should get a very straight line, which will make it easier to accurately interpolate the value of P for L = 0.25 m.
("sqrt" means "square root of").
b) Assume you are trying to match the period for a L = 0.25 m pendulum. Get the value of that period from (a). The, for a spring, set
P = 2 pi sqrt (m/k)
Solve for the spring constant, k.
A helical spring is 59 cm long when a load of 100 N is hung from it and 62 cm long when the load is 120 N. Find its spring constant.
a) To determine the period of a pendulum with a length of 0.25m, we can interpolate using the data given. First, plot the graph of Period (P) vs Length (L) using the given data:
Length (m) Period (s)
0.05 0.30
0.20 0.90
0.40 1.30
0.60 1.60
0.80 1.80
1.00 2.00
Now, plot the data points on a graph with Period (P) on the y-axis and Length (L) on the x-axis. Connect the points with a smooth, continuous line.
Next, calculate the square root of the Length (L) values and plot them on a new graph, with Period (P) still on the y-axis. Plot the square root of the Length (L) values against their corresponding Period (P) values. You should observe a very straight line on this graph.
Now, interpolate the value of Period (P) for L = 0.25m by finding the point on the line that corresponds to L = 0.25m. Read the corresponding Period (P) value from the y-axis.
b) To determine the spring constant (k) that would result in the same period as the pendulum with L = 0.25m, we use the equation:
P = 2π√(m/k)
Rearranging the equation:
k = (4π^2m) / P^2
Plug in the mass (m) of the pendulum, and the Period (P) value for L = 0.25m that you obtained in part a. Solve for k to get the value of the spring constant that would result in the same period as the pendulum.