Four equal, uniform rods of mass m and length 2a are hinged together to form a rhombus ABCD. The point A is fixed, while C lies directly beneath it and is free to slide up and down. The whole system can rotate around the vertical. Let 0 be the angle that AB makes with the vertical, and o be the angular velocity around the vertical, as shown in figure 3. Find the Lagrangian for this system and show that there are two conserved constants of motion. B Figure 3: The rotating rhombus.

ms mp mL A small frog of mass me clings to the left end of a uniform pole of mass mp and length L, and a large frog of mass my clings to the right end of the pole as shown in this top-down view. The system is rotating about its center of mass. The large frog is moving with tangential speed UL. Treat the frogs as point masses. Show all work and solve in terms of given variables. A. Find the center of mass of the system measured from the right end of the pole. B. What is the magnitude of the angular velocity of the system?

4. Find the velocities of points B, C, and D of the double-slider mechanism illustrated in the figure below if crank 2 rotates at 42 rad/s cw. Use the graphical method.... y -3". 5 160 D 120° 3 Figure P3.22 RAO₂ = 2 in, RBA = 10 in, RCA=4 in, RCB = 7 in, and RDC = 8 in.

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A solid cylinder (mass 1.2kg, radius 5.0cm) on a horizontal table is connected to a block (mass 200 g) via a pulley (mass 0.80kg, radius 10.0cm) via a light, rigid string. Draw free body diagram for each of the object 1) list the dynamic and kinematic equations that related to all three objects. 2) find the acceleration of the falling block 3) fnd the angular acceleration of the cylinder and the friction force on it

A mechanical system consists of a ("long") rigid rod of length d and mass M, and another ("short") rigid rod of length I and mass m. The rods are attached at a fixed 90° angle such that the endpoint of the long rod touches the middle of the short rod; see the figure below. The mass per unit length of the rods is constant. The system can move in a vertical plane (x, y) with the endpoint O of the long rod fixed, so that the system is free to oscillate in the vertical plane about the point O. Gravity acts along the vertical y direction. 0 d a) How many degrees of freedom does the system have? [Expect to use a line to answer this question.] â b) Calculate the moment of inertia of the system constituted by the rods about an axis orthog- onal to the plane (x, y) and passing through O. Assume that the widths of the rods are negligible, so that you can effectively treat them as one-dimensional objects. [Expect to use about half a page to answer this question.] c) Determine the distance of the…

A rod assembly rotates around the z-axis. The mass C is 10 kg and its initial velocity is 7 m/s. A moment and force both acts as shown in the figure where M = 8t2+5  N.m and F = 60 N. Find the velocity of mass C after 2 seconds.

Solve accordingly to given 3 parts below

Consider a satellite moving in a torque-free environment. The inertial frame and satellite body-fixed frame are represented by N-frame and B-frame, where {^₁, ŵ2, ñ3} and {b₁, 62, 63} are right-handed vector bases fixed in N-frame and B-frame, respectively. {1;} is aligned with the principal axis of inertia, where ☎₁ and 3 are associated with the minimum and maximum moments of inertia, respectively. Denote the satellite angular velocity by w = wib; and the inertia tensor about the CoM by Ic. Suppose that I at time t = 0 is given by: Ic(t = 0) = 2ñ₁ñ₁ +2.25 ñ¿Ñ₂ +2.25 ñ³Ñ3 – 0.75 ñ¿ñ3 −0.75 3₂ (dimensionless) Express the satellite inertia tensor about the CoM in B-frame. Determine [BN] at time t = 0, i.c., the direction cosine matrix (DCM) that represents the orientation of B-frame relative to N-frame.