Physics 110 Lab 9 - Experiment 7: Hanging Masses and Pendulums PDF

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Experiment 7: Hanging Masses and Pendulums...

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Experiment 7: Hanging Masses and Pendulums! Pre-Lab Reading Discussion! Hanging masses on a spring and pendulums both exhibit periodic motion, both due to the presence of their respective restoring forces. In the absence of friction/drag forces, and for small pendulum displacements, this motion is known as Simple Harmonic Motion (SHM).!

For a mass on a spring, the restoring force of the spring on the mass is linearly proportional to the displacement of the mass from equilibrium, known as Hooke’s Law:!

F = − k x, or F = k x Where x is the displacement from equilibrium, and k is known as the spring constant, or stiffness of the spring. In SI units, k is measured in N/m. Under SHM, the motion x(t) of the mass m is:!

x(t) = X0cos(ωt) Where X0 is the initial displacement, and ω

=

k is the frequency of oscillation in rad/s. The m

T=

2π ω

period of oscillation is also defined as: !

!"#"$%&'()*+",-"!2 = 4π2M/k! We will use the form 4π2M = kT2 + 0! 4π2M(T2) = kT2 + 0,! Y(x) = mx + b ! a linear equation where the physical meaning of the slope is k, the stiffness of the spring.! 4π2M(T2) = kT2 + 0 is a linear equation that depends on T2, therefore it is not a quadradic equation. You must square T first then find the dependent variable 4π2M. This form of the equation states that the constant 4π2 times the mass hanging on a spring is determined by the square of its period times the stiffness of the spring. It’s for mathematical convenience because you will want to determine the slope of the linear plot of M vs T2, which is k.!

Similarly, a pendulum displaced from equilibrium will swing back and forth through both the force of gravity on the mass and the tension in the string holding the mass. If the motion is restricted to small angular displacements θ , then the pendulum also undergoes SHM:!

θ(t) = θ0cos(ωt ) Where θ0 is the initial angular displacement, and ω

rad/s. Then its period of oscillation is T = 2 π

=

g is the frequency of oscillation in l

L , with the help of algebra, the equation for g

the L vs T2! 4π2L(T2) = gT2 + 0, g = 9.8m/s2, L is the length of the string! This time the physical meaning of the slope is g, acceleration due to gravity.! Pre-Lab Question! ●

Prove that when a mass M hanging on a spring is not moving at all, prove that

Mg = kΔx where Δx is the displacement of the spring. Use Newton’s Second Law and the Free Body Diagram. !

Procedure – Hanging Mass! 1. Open the simulation called ‘Masses and Springs’, found here. Navigate to the “Lab’ tab.! h"ps://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html!

!

2. Force versus displacement: Take the 100g mass, and attach it to the spring. The spring/ mass combo should move back and forth. In the absence of friction or damping, the spring will oscillate forever. Using the damping option, damp the system a little (or a lot) to get the new system to rest, and measure the new equilibrium position. a. If you cannot get the damping to work, an alternative option would be to show the Mass equilibrium and Natural Length lines on the top right, and measure the difference using the ruler. b. The 100g mass can be varied using the slider on the top. Repeat this measurement for at 5 values of the mass. 3. Plot the Force versus displacement data for the hanging mass. Include screenshots of the simulation to support your data. Calculate analytically and graphically the force constant of the spring and compare the two values. The analytic calculation ! involves using Hooke’s Law for each data point. 4. Period vs Mass: This time, we will measure the period T of oscillation by attaching a mass to the spring and measuring time with a stopwatch. As it can be difficult to measure the period of one single oscillation, the recommended strategy is to record the time it takes to complete 25 oscillations (complete revolutions or complete back and forths) and finding the average time per oscillation. a. Do this for the 50g, 100g, 200g, and 300g masses.

Analysis – Hanging Mass! A. Plot the Force versus displacement data for the hanging mass. Include screenshots of the simulation to support your data. Calculate analytically and graphically the force constant of the spring and compare the two values. The analytic calculation ! involves using Hooke’s Law for each data point.!

B. Plot T vs M. Does the graph make physical sense? !

C. Plot T2 vs M. . Include screenshots of the simulation to support your data. How does this graph differ from B? Explain.!

a. For this graph, find graphically the spring constant k.!

D. Compare the values of k found in both A and D. By what percent do they differ?!

E. Does the period of SHM depend on the amplitude? How well does your data justify your answer?!

Procedure – Pendulum! 5. Now, open the simulation called ‘Pendulum’, found here. Navigate to the ‘Lab’ tab.! h"ps://phet.colorado.edu/sims/html/pendulum-lab/latest/pendulum-lab_en.html!

!

6. Period versus length: The pendulum begins with a 1.00kg mass on it. Displace the pendulum a small amount (no more than 10°) and count (using the stopwatch) the amount of time it takes to complete 25 oscillations. Find the period of oscillation using this measurement.! a. Repeat this measurement for 3 other lengths: 0.5m, 1.0m, and 0.2m.! b. Now, change the length back to 0.7m. Play around with the masses and notice how this may (or may not) change the period of oscillation! 7. Repeat 5 and 5a for one other value of gravity. You may choose the Moon, Jupiter, or a custom value of gravity (so long as you specify what value g is)!

Data Tables! Pendulum – Period versus Length! Run 1: Earth gravity! g = _______________! Mass (kg)

Length (m)

Time for 50 oscillations (s)

Time per oscillation (s)

Time for 50 oscillations (s)

Time per oscillation (s)

Run 2: Custom gravity! g = _______________! Mass (kg)

Length (m)

Analysis – Pendulum! A. Plot your L vs T2 = gT2 + 0, 4π2L(T2) = gT2 + 0, !

B. data for the pendulum on earth and your other planet.Include screenshots of the simulation to support your data. According to the theory, what should the slope of this line be physically?!

C. Calculate the value of g for each set of data and compare to the known values of g.!

Post-lab questions (Be sure to show your work)! 1. What is the spring constant of a spring that needs a force of 7N to be compressed from 50 cm to 42cm? Find k.!

2. A vertical spring with stiffness k originally is at rest with no mass attached. Then, a mass M is attached, and the spring rocks back and forth until it is at rest at a new equilibrium position. What is this position in terms of g, M, and k?!

3. Grandfather clocks are just large pendulums with a period of two seconds, so each end of the oscillation corresponds to one second passing. What must the length of the pendulum be to achieve this?!...