Question

Notify Me Bookmark

Consider the spring-mass system shown below, where the mass is subjected to the gravitational force mg. (a) Represent the free-body diagram needed to derive the equation of motion. Identify the magnitude and direction of all forces. (b) Using Newton's second law, derive the equation of motion as a linear, time-invariant ODE. (c) Using the differentiation theorem, find the Laplace transform of the equation of motion, and express the function Y(s) = L[y(t)]. Assume initial conditions y(0) = 1 and y˙(0) = 0.

Consider the spring-mass system shown below, where the mass is subjected to the gravitational force mg. (a) Represent the free-body diagram needed to derive the equation of motion. Identify the magnitude and direction of all forces. (b) Using Newton's second law, derive the equation of motion as a linear, time-invariant ODE. (c) Using the differentiation theorem, find the Laplace transform of the equation of motion, and express the function Y(s) = L[y(t)]. Assume initial conditions y(0) = 1 and y˙(0) = 0.

Image text
Problem 1 (10 points): Consider the spring-mass system shown below, where the mass is subjected to the gravitational force m g . (a) Represent the free-body diagram needed to derive the equation of motion. Identify the magnitude and direction of all forces. (b) Using Newton's second law, derive the equation of motion as a linear, time-invariant ODE. (c) Using the differentiation theorem, find the Laplace transform of the equation of motion, and express the function Y ( s ) = L [ y ( t ) ] . Assume initial conditions y ( 0 ) = 1 and y ˙ ( 0 ) = 0 .

Detailed Answer

0 0

Please enter your email here. You will get email when this question is answered by our tutor.

Doubtrix Logo

Problem is not yet solved!

Click on Notify me to get email when question is answered.

Join Doubtrix with 7-days free trial

  • Icon Correct & Verified Answers
  • Icon No AI-Generated Answers
  • Icon Video Solution for Difficult Questions
  • Icon Work With Experts to Reach at Correct Answers
Notify me Login Sign Up

Similar/Related Questions

Consider two blocks M1 on top of M2 on a level table. There is no friction between block M2 and the table, but there is friction between block M1 and block M2. Denote the coefficients of static and kinetic friction between the blocks as μs and μk, respectively. The applied force F is applied to block M1 at an angle θ below the horizontal. The blocks start from rest. First, we will consider the situation where F is small enough that the two blocks stick together. (a) Draw free-body diagrams for the two masses. Be sure to label all forces and the direction of the accelerations. Also be sure to define an appropriate coordinate system for each free-body diagram. (b) Write down the Newton's second law equations for each block and each dimension (x and y). (c) Write down an inequality relating the force of friction and the normal force. (d) Identify the two Newton's third law pairs - i. e. the pairs of forces between the two blocks that are equal and opposite. (e) Find the acceleration of the blocks as a function of F, θ, M1, M2, g, μs, and μk. (You don't have to use all of these, but you will use at most these. ) (f) Find a condition on F such that the blocks stick together. i. e. Write an inequality in the form F ≤ Fmax for the blocks to stick together where Fmax is in terms of θ, M1, M2, g, μs, and μk.
Solution
0 0

A box is on a rough surface that is inclined at an angle of 30.0∘ to the horizontal. A mass, m1 = 0.270 kg, is attached to a hanging mass of m2 = 105 g over an ideal pulley. a. Draw two free-body diagrams, one for each mass, showing all forces acting on them and chosen direction of motion. b. What is the value of μ that allows the block to slide down the incline at constant speed? Blank 1 c. The incline is then greased to remove the friction and the masses released again from rest. Find the tension in the string, in Newtons, Blank 2 and the acceleration of the box, in m/s2. Blank 3 d. How long, in seconds, does it take the box to slide a distance of 1.40 m along the incline? Blank 4 For this problem, SHOW ALL WORK including a free-body diagram of forces on each mass shown in the drawing below. The direction of accelerations should be clearly indicated on the sketch. Show all equations used before entering numbers. This problem should be well-structured. DO NOT ENTER UNITS WITH ANSWERS.
Solution
0 0

A beam resting on two pivots has a length of L = 6.00 m and mass M = 91.0 kg. The pivot under the left end exerts a normal force n1 on the beam, and the second pivot placed a distance ℓ = 4.00 m from the left end exerts a normal force n2. A woman of mass m = 58.5 kg steps onto the left end of the beam and begins walking to the right as in the figure below. The goal is to find the woman's position when the beam begins to tip. (i) (a) Sketch a free-body diagram, labeling the gravitational and normal forces acting on the beam and placing the woman x meters to the right of the first pivot, which is the origin. (Submit a file with a maximum size of 1 MB.) Choose file no file selected This answer has not been graded yet. (b) Where is the woman when the normal force n1 is the greatest? x = m (c) What is n1 when the beam is about to tip? N (d) Use the force equation of equilibrium to find the value of n2 when the beam is about to tip. N (e) Using the result of part (c) and the torque equilibrium equation, with torques computed around the second pivot point, find the woman's position when the beam is about to tip. x = m (f) Check the answer to part (e) by computing torques around the first pivot point. x = m Except for possible slight differences due to rounding, is the answer the same? Yes No
Solution
0 0