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Exercise 1: Rolling cylinder A homogenous cylinder is rolling down a plane (inclination a towards the horizontal plane) without sliding. The moment of inertia of the cylinder for rotations about its symmetry axis is given by I = mr²/2. Find the equations of motion.

Exercise 1: Rolling cylinder A homogenous cylinder is rolling down a plane (inclination a towards the horizontal plane) without sliding. The moment of inertia of the cylinder for rotations about its symmetry axis is given by I = mr²/2. Find the equations of motion.

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Exercise 3: Sliding rope A rope of length l and mass p per length unit slides down over the edge of a table after being released without friction (cp. Figure). Solve the equation of motion for the initial values: r(0) = a with a <l and i(0) = 0. Exercise 4: Unrolling Cylinders homogenous cylinders tia I = mırf/2, 12 = mrf/2 are wrapped by a thread (see figure). The first mass m1, m2: ri, ra and mome iner- cylinder is rotating on a fixed axis without friction. The second cylinder is falling down vertically in the gravitational field. Derive the equations calculate the thread tension. motion and x-Achse Exercise 5: Bead on rotating wire On a parabola-like wire (cp Example 2 in the lectures) a bead is moving un- der the constraint z = ar? (in cylindrical coordinates). The wire rotates with constant angular velocity w about the z-axis. Gravity is acting in the negative z-direction. a) Find the Lagrange function L and show that for w = V2ag the sum of centrifugal and gravitational force is perpendicular to the wire. b) Solve the equation of motion.
Transcribed Image Text:Exercise 3: Sliding rope A rope of length l and mass p per length unit slides down over the edge of a table after being released without friction (cp. Figure). Solve the equation of motion for the initial values: r(0) = a with a <l and i(0) = 0. Exercise 4: Unrolling Cylinders homogenous cylinders tia I = mırf/2, 12 = mrf/2 are wrapped by a thread (see figure). The first mass m1, m2: ri, ra and mome iner- cylinder is rotating on a fixed axis without friction. The second cylinder is falling down vertically in the gravitational field. Derive the equations calculate the thread tension. motion and x-Achse Exercise 5: Bead on rotating wire On a parabola-like wire (cp Example 2 in the lectures) a bead is moving un- der the constraint z = ar? (in cylindrical coordinates). The wire rotates with constant angular velocity w about the z-axis. Gravity is acting in the negative z-direction. a) Find the Lagrange function L and show that for w = V2ag the sum of centrifugal and gravitational force is perpendicular to the wire. b) Solve the equation of motion.
Kohrad Dr Anirudh Segireddy Exercises for Classical Mechanics 214 Second semester 2020 Exercise sheet 2, Tutorials 27 Nov 2020 Exercise 1: Rolling cylinder A homogenous cylinder is rolling down a plane (inclination a towards the horizontal plane) without sliding. The moment of inertia of the cylinder for rotations about its symmetry axis is given by I = mr2/2. Find the equations of motion. Exercise 2: Planar Double Pendulum Write down the equation of motion using Lagrange's equations of second kind for a planar double pendulum (cp Figure), where a second pendulum is suspended from the first pendulum. Exercise 3: Sliding rope A rope of length l and mass p per length unit slides down over the edge of a table after being released without friction (cp. Figure). Solve the equation of motion for the initial values: r(0) = a with a <l and i(0) = 0. Exercise 4: Unrolling Cylinders Two homogenous cylinders of mass m1, m2, Radii ri, r2 and moments of iner- tia I = mır{/2, I2 = mır{/2 are wrapped by a thread (see figure). The first cylinder is rotating on a fixed axis without friction. The second cylinder is falling down vertically in the gravitational field. calculate the thread tension. Derive the equations of motion and x-Achse P2 Exercise 5: Bead on rotating wire
Transcribed Image Text:Kohrad Dr Anirudh Segireddy Exercises for Classical Mechanics 214 Second semester 2020 Exercise sheet 2, Tutorials 27 Nov 2020 Exercise 1: Rolling cylinder A homogenous cylinder is rolling down a plane (inclination a towards the horizontal plane) without sliding. The moment of inertia of the cylinder for rotations about its symmetry axis is given by I = mr2/2. Find the equations of motion. Exercise 2: Planar Double Pendulum Write down the equation of motion using Lagrange's equations of second kind for a planar double pendulum (cp Figure), where a second pendulum is suspended from the first pendulum. Exercise 3: Sliding rope A rope of length l and mass p per length unit slides down over the edge of a table after being released without friction (cp. Figure). Solve the equation of motion for the initial values: r(0) = a with a <l and i(0) = 0. Exercise 4: Unrolling Cylinders Two homogenous cylinders of mass m1, m2, Radii ri, r2 and moments of iner- tia I = mır{/2, I2 = mır{/2 are wrapped by a thread (see figure). The first cylinder is rotating on a fixed axis without friction. The second cylinder is falling down vertically in the gravitational field. calculate the thread tension. Derive the equations of motion and x-Achse P2 Exercise 5: Bead on rotating wire
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