Preliminary Dynamics

Note: 

1. Show details of all steps in your HW.
2. HW’s are intended to be individual assignments. So, in the spirit of Widener guidelines on academic integrity and honesty, your HW must provide sufficient evidence that this is your original work.

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Problem 1) Consider the planar pendulum kinematics discussed in class, start with position vector 𝑟^𝑃/𝑂 resolved in terms of the unit vectors 𝑖 and 𝑗, and verify the expressions obtained for the acceleration and velocity:

𝑎^𝑃/𝑂 = (𝐿𝜃̈ cos 𝜃 − 𝐿𝜃̇² sin 𝜃)𝑖 + (𝐿𝜃̈ sin 𝜃 + 𝐿𝜃̇² cos 𝜃 )𝑗

Problem 2) Consider the kinematic of the rolling disc considered in class, and verify that the instantaneous acceleration of the point of contact is not zero.

Problem 3) In Figure below, a slide of mass 𝑀_𝑟 is located on a bar whose angular displacement in the plane is described by the coordinate 𝜃. The motion of the slide from the pivot point is measured by the coordinate 𝑟₁. The acceleration due to gravity acts in a direction normal to the plane of the motion. Assume that the point 𝑂 is fixed in an inertial reference frame and determine the absolute velocity and absolute acceleration of the slide.

Problem 4) Determine the number of degrees of freedom of the systems shown below. Assume that the length L of the pendulum shown in (a) is constant and that the length between each pair of particles in b is constant.

Hint: For c, the rigid body can be thought of as a system of particles where the length between each pair of particles is constant.