Question

1. The structure shown here is made from six pieces attached as shown. ring ro Each piece has the same mass, M. The pieces are as follows One solid, uniform disk with diameter D Four identical thin uniform rods of length D, connected to the disk as shown (opposing pairs of rods aligned with the disk center) One thin, uniform ring connected to the outer ends of the four rods, as shown The entire structure is laid flat, at rest, on a frictionless surface and pinned to the surface with a frictionless pin at Point P. (So the diagram is an overhead view.) Point P is located at the midpoint of one of the rods, as shown. Point X is located on the edge of the disk- directly opposite to point P-as shown ro disk rod ro Then at time t = 0, a steady torque, τ, is applied to the structure around point P, and as a result, the structure begins to rotate around that pin a. Find an algebraic expression for a(t), the magnitude of the acceleration of point X, as a function of time, t b. Find an algebraic expression for W(t), the total work done on the structure, as a function of time, t Each expression may contain only known values (and the variable ). These are known values: M, D, T,

Answer 1

The structure shown hero is made from six pieces attached as shown Eeach pieace has the mass m the acceleration has 2 parts. I is tangential and the other is interptial and the other is centripetal a_T = 3d/2 alpha a_x = 3d/2 w^2 Thus, the inagrilutie of acc is a_x = squareroot a_T^2 + a_0^2 = 3d/2 squareroot alpha^2 + w^4 = 3d/2 squareroot T^2/I_p^2 + (it/Ip)^4 = 3d/2Ip squareroot 1 + I^2 t^4/I_p^2 (b) Total work done \r\n (b) w(t) = integral^t_0 Tw dt implies w(t) = I^2 t^2/2I_p Ip is the moment of inertia (MOI) about axis at p First find the MOI about the centre of mass (COM) I_COM = 1/2 M(D/2)^2 + 4 times (MD^2 + 1/12 MD^2) + M(30/2)^2 = 161/24 MD^2 Parallel axis the ern Ip = I_COM + 6MD^2 IP = 305/4 MD^2