You can make an interesting pendulum by taking a very light, stiff, cylindrical object (like a paper towel roll) of length \(L\) and taping or gluing two small, heavy objects to the inside: mass \(m_1\) at an end and mass \(m_2\) exactly in the center. The end with nothing attached to it is then attached to the ceiling with a very short piece of string, and the pendulum is allowed to swing back and forth with an amplitude much smaller than \(L\text{.}\)
In case A, suppose that \(m_1 \gt m_2\text{,}\) and in case B, suppose that \(m_2 \gt m_1\text{.}\) Will the period in case A be greater than, less than, or equal to the period in case B?
A string (length \(L\)) with constant tension \(T\) is tied horizontally between two walls (this is effectively how strings in many musical instruments work!). An ant (mass \(m\)) clings to the very center of the string, which is moving back and forth horizontally with amplitude \(A\) much smaller than \(L\text{.}\)
Model the situation as a simple harmonic oscillator, find the angular frequency, and create one or more graphs showing the motion of the system as a function of time.
A mass is compressed between two springs on a frictionless horizontal surface. When the mass is in equilibrium, both springs are at their relaxed length. Which of the following statements are true regarding this situation?
A mass on an ideal spring and a simple pendulum have the same oscillation frequency on Earth. On Jupiter’s moon Europa, the gravitational acceleration is about 1/9 that of Earth’s. How would the frequency of the pendulum compare to the frequency of the mass-spring system if they were both oscillating on Europa?
On a windless day, Callie the Cardinal-Grosbeak (mass = 43 g) is swaying on a tree twig of negligible mass. Being quite the scientist, she notices that the tree branch behaves just like a spring; she records her position as being described by this time function: \(x(t) = (5 cm) cos(\frac{4\pi}{3}t)\text{.}\) What is the maximum acceleration Callie experiences?
A mass is connected to a spring and is set to oscillating. You start your stopwatch just as the mass passes the equilibrium position, traveling in negative x-direction. Which time function should be used to model the position of the mass?
