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A uniform hoop of mass m and radius R rolls down an inclined plane of length I without slipping. The plane, which is originally horizontal, is lifted up at a constant rate such that the angle of the plane with the horizontal at time t is o = wt (see figure). R g O = 0 t (a) i. q and 0 are not independent. If 0 = 0 when q = 1, determine the relation between them. ii. Determine the height of the centre of mass of the hoop above the base of the plane. ii. Hence show that the Lagrangian of the body, expressed in terms of the distance q from the pivotal point of the plane, is L = mở² +mu q – mg(q sin wt + Rcos wt).

A uniform hoop of mass m and radius R rolls down an inclined plane of length I without slipping. The plane, which is originally horizontal, is lifted up at a constant rate such that the angle of the plane with the horizontal at time t is o = wt (see figure). R g O = 0 t (a) i. q and 0 are not independent. If 0 = 0 when q = 1, determine the relation between them. ii. Determine the height of the centre of mass of the hoop above the base of the plane. ii. Hence show that the Lagrangian of the body, expressed in terms of the distance q from the pivotal point of the plane, is L = mở² +mu q – mg(q sin wt + Rcos wt).

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A uniform hoop of mass m and radius R rolls down an inclined plane of length I without slipping. The plane, which is originally horizontal, is lifted up at a constant rate such that the angle of the plane with the horizontal at time t is o = wt (see figure). R g O = 0 t (a) i. q and 0 are not independent. If 0 = 0 when q = 1, determine the relation between them. ii. Determine the height of the centre of mass of the hoop above the base of the plane. iii. Hence show that the Lagrangian of the body, expressed in terms of the distance q from the pivotal point of the plane, is L = ma² +ms'q? – mg(a sin wt + Rcos wt). (b) Determine the Euler-Lagrange equations for the system. Note: you may assume the Lagrangian equations of motion for generalised co-ordinates q:: d (OL
Transcribed Image Text:A uniform hoop of mass m and radius R rolls down an inclined plane of length I without slipping. The plane, which is originally horizontal, is lifted up at a constant rate such that the angle of the plane with the horizontal at time t is o = wt (see figure). R g O = 0 t (a) i. q and 0 are not independent. If 0 = 0 when q = 1, determine the relation between them. ii. Determine the height of the centre of mass of the hoop above the base of the plane. iii. Hence show that the Lagrangian of the body, expressed in terms of the distance q from the pivotal point of the plane, is L = ma² +ms'q? – mg(a sin wt + Rcos wt). (b) Determine the Euler-Lagrange equations for the system. Note: you may assume the Lagrangian equations of motion for generalised co-ordinates q:: d (OL
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