Consider a pendulum with a bob of mass m that is suspended from a spring rather...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Consider a pendulum with a bob of mass m that is suspended from a spring rather than a rigid rod. Assume that the spring has spring constant k, that it has negligible rest length and mass (that is, you can treat both as zero), that the spring only stretches (it does not bend), and that the pendulum oscillates in a plane (making this a two-dimensional problem). The force due to the spring follows Hooke's law, F =-kr (and there is of course also the gravitational force, with magnitude g, pointing downward). a) (3 points) Write down Newton's second law for this system in Cartesian coordinates in the form of two differential equations for x(t) and y(t). You do not need to solve the differential equations. However, be sure to draw a free-body diagram in which you also define and label your coordinates. b) (4 points) Repeat a), but now in polar coordinates (r, o). Again, you do not need to solve the differential equations for r(t) and o(t), but do draw a free-body diagram and label your coordinates. You also do not need to simplify or decouple the equations; just leave them as one equation related to F, and another related to Fo. c) (4 points) Verify (by substitution) that x(t) = Ax cos(wt + 6r) and y(t) = Ay cos(wt + y) +h satisfy the differential equations that you found in part a) for arbitrary real constants Ar Ay, dr, and dy, and specific constant values of w and h. Determine the values of w and h in terms of the other constants (g, k, and m) of this problem. d) (3 points) Suppose that r = = 0 (which corresponds to no initial velocity) and (0) = y(0)=h/2. What is r(t), the distance from the origin as a function of time? [Hint: Don't attempt to solve the differential equation in part b); try to obtain it from your solution in part c) when specialized to the initial conditions given in this part.] Consider a pendulum with a bob of mass m that is suspended from a spring rather than a rigid rod. Assume that the spring has spring constant k, that it has negligible rest length and mass (that is, you can treat both as zero), that the spring only stretches (it does not bend), and that the pendulum oscillates in a plane (making this a two-dimensional problem). The force due to the spring follows Hooke's law, F =-kr (and there is of course also the gravitational force, with magnitude g, pointing downward). a) (3 points) Write down Newton's second law for this system in Cartesian coordinates in the form of two differential equations for x(t) and y(t). You do not need to solve the differential equations. However, be sure to draw a free-body diagram in which you also define and label your coordinates. b) (4 points) Repeat a), but now in polar coordinates (r, o). Again, you do not need to solve the differential equations for r(t) and o(t), but do draw a free-body diagram and label your coordinates. You also do not need to simplify or decouple the equations; just leave them as one equation related to F, and another related to Fo. c) (4 points) Verify (by substitution) that x(t) = Ax cos(wt + 6r) and y(t) = Ay cos(wt + y) +h satisfy the differential equations that you found in part a) for arbitrary real constants Ar Ay, dr, and dy, and specific constant values of w and h. Determine the values of w and h in terms of the other constants (g, k, and m) of this problem. d) (3 points) Suppose that r = = 0 (which corresponds to no initial velocity) and (0) = y(0)=h/2. What is r(t), the distance from the origin as a function of time? [Hint: Don't attempt to solve the differential equation in part b); try to obtain it from your solution in part c) when specialized to the initial conditions given in this part.]
Expert Answer:
Related Book For
Physics
ISBN: 978-0077339685
2nd edition
Authors: Alan Giambattista, Betty Richardson, Robert Richardson
Posted Date:
Students also viewed these physics questions
-
CANMNMM January of this year. (a) Each item will be held in a record. Describe all the data structures that must refer to these records to implement the required functionality. Describe all the...
-
Let A, B be sets. Define: (a) the Cartesian product (A B) (b) the set of relations R between A and B (c) the identity relation A on the set A [3 marks] Suppose S, T are relations between A and B, and...
-
Solve: [1+ log (xy)] dx + [ 1 + x/y ]dy = 0.
-
An urn contains one red chip and one white chip. One chip is drawn at random. If the chip selected is red, that chip together with two additional red chips are put back into the urn. If a white chip...
-
The random variable X follows a Poisson process with l = 0.05 and t = 30. Find each of the following: (a) P(2) (b) P(X < 2) (c) P(X 2) (d) P(1 X 3) (e) What are X and X?
-
Summarize different theories offered to explain the existence of gender hierarchies.
-
Selected year-end financial statements of Overton Corporation follow. (All sales were on credit; selected balance sheet amounts at December 31, 2012, were inventory, $17,400; total assets, $94,900;...
-
1. Evergreen company is investigating the feasibility of buying a new production line producing a new product. They project unit sales as in the below table, and they project price per unit to be...
-
20 The arm BC is attached to the sliding block and is made to rotate counterclockwise at a constant rate of 2 rev/min. Determine the acceleration as of the block C and the angular acceleration as of...
-
Which one of the following statements is true? a) The South is the most heavily unionized section of the country. b) There are very few right-to-work states in the South. c) New York is the most...
-
Deregulation of the trucking industry resulted in __________. a) many more firms and lower prices b) many more firms and higher prices c) fewer firms and lower prices d) fewer firms and higher prices
-
Repeat Problem 6.6 using the Gauss-Seidel iterative method. Which method converges more rapidly? Problem 6.6 Using the Jacobi iterative method. Start with \(x_{1}(0)=\) \(x_{2}(0)=x_{3}(0)=0\), and...
-
__________________ is the main arena of the power struggle between labor and management.
-
If the minimum wage were eliminated, the employment of marginal workers would _________. a) rise a lot b) rise a little c) stay exactly the same d) fall a little e) fall a lot f) fall by an...
-
Read the article The beauty myth and Why nice guys finish last and answer that how did these readings resonates with you or your experiences on a personal level? In what ways did they allow you to...
-
In Exercises 1-2, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. 1. The logarithm of the difference of two numbers is equal to...
-
Describe the situation for which but-for or collapsed as-built analysis is most appropriate.
-
The entire process of project time management can be enhanced considerably by scheduling periodic job progress meetings. What is the objective of such a meeting, and who should participate?
-
Why is an independent review of work accomplishment preferred over having the foreman responsible for the work do the assessment?
Study smarter with the SolutionInn App