QuestionQuestion

Transcribed TextTranscribed Text

A uniform ladder of length l and mass m has one end on a smooth horizontal floor and the other end against a smooth vertical wall. The ladder is initially at rest and makes an angle 00 with respect to the horizontal. y (x,y) o X Figure 8.5: A ladder sliding down a wall and across a floor. (a) Make a convenient choice of generalized coordinates and find the Lagrangian. Solution : I choose as generalized coordinates the Cartesian coordinates (x,y) of the ladder's center of mass, and the angle 0 it makes with respect to the floor. The Lagrangian is then L 1m(i2+y)+1102+mgy There are two constraints: one enforcing contact along the wall, and the other enfore- ing contact along the floor. These are written (b) Prove that the ladder leaves the wall when its upper end has fallen to a The equations of motion are Thus, we have mji + mg=, 10= - = Qg. We now implement the constraints to eliminate I and y in terms of 0. We have 7 We can now obtain the forces of constraint in terms of the function @(t): A1 = A2 = 1 mm + mg - We substitute these into the last equation of motion to obtain the result or with I This may be integrated once (multiply by é to convert to a total derivative) to yield 0 = 2w3 sin to which is of course a statement of energy conservation. This, We may now obtain () and Demanding A1 (0) = 0 gives the detachment angle 0 = ea where Note that 2(d) = mgo/(1 + a) > 0, so the normal force from the floor is always positive for 0>Bd The time to detachment is now + sin

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:
    Solution.pdf.

    $8.00
    for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Classical Mechanics Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats