Oscillating Test Tube
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
A test tube is immersed in a liquid to a height h_. After a slight push it starts to oscillate about its equilibrium position. abcliste abc Show that the motion of the test tube is a simple harmonic motion. Derive a formal expression for the angular frequency. abc Calculate the oscillation period for a test tube with diameter dO and mass mO in water. abcliste
Solution:
abcliste abc In the equilibrium postion the gravitational force F_G is cancelled by the buoyancy force F_B: F_G F_B mg rho V g rho d/^pi h_ g When displaced from the equlibrium position by a vertical distance yt the gravitatoinal force remains the same but buoyancy is changed by Delta F_B -rho d/^pi Delta h g There is a minus sign because a displacement in the positive upward direction leads to a reduction in buoyancy. With sscFrestDelta F_B and ytDelta h we can rewrite this as sscFrest -rhod/^pi g yt m ddotyt -rhod/^pi g yt This is the characteristic differential for a simple harmonic motion with an angular frequency given by omega^ fracrhod/^pi gm fracrho d^pi gm i.e. omega fracdsqrtfracrho pi gm abc It follows for the oscillation period T fracpiomega TF fracpidtimessqrtfracmRWatimespitimesncg T approx resultTP abcliste
A test tube is immersed in a liquid to a height h_. After a slight push it starts to oscillate about its equilibrium position. abcliste abc Show that the motion of the test tube is a simple harmonic motion. Derive a formal expression for the angular frequency. abc Calculate the oscillation period for a test tube with diameter dO and mass mO in water. abcliste
Solution:
abcliste abc In the equilibrium postion the gravitational force F_G is cancelled by the buoyancy force F_B: F_G F_B mg rho V g rho d/^pi h_ g When displaced from the equlibrium position by a vertical distance yt the gravitatoinal force remains the same but buoyancy is changed by Delta F_B -rho d/^pi Delta h g There is a minus sign because a displacement in the positive upward direction leads to a reduction in buoyancy. With sscFrestDelta F_B and ytDelta h we can rewrite this as sscFrest -rhod/^pi g yt m ddotyt -rhod/^pi g yt This is the characteristic differential for a simple harmonic motion with an angular frequency given by omega^ fracrhod/^pi gm fracrho d^pi gm i.e. omega fracdsqrtfracrho pi gm abc It follows for the oscillation period T fracpiomega TF fracpidtimessqrtfracmRWatimespitimesncg T approx resultTP abcliste