Rotational MotionHard
Question
Three particles, each of mass m are placed at the points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) on the inner surface of a paraboloid of revolution obtained by rotating the parabola, x2 = 4ay about the y-axis. Neglect the mass of the paraboloid. (y-axis is along the vertical)
Options
A.The moment of inertia of the system about the axis of the paraboloid is I = 4ma (y1 + y2 +y3).
B.If potential energy at O is taken to be zero, the potential energy of the system ismg (y1 + y2 + y3)
C.If the particle at (x1, y1, z1) slides down the smooth surface, its speed at O is 
D.If the paraboloid spins about OY with an angular speed ω the kinetic energy of the system will be ma (y1 + y2 + y3)ω2
Solution
For any point on the surface of paraboloid, (x2 + z2) = 4ay
(A) I =
m(x12 + z12) (distance of mi from y-axis is 
) = 4ma (y1 + y2 + y3).
(B) mg (x1 + x2 + x3) = mg(y1 + y2 + y3)
(C) mg x1 =
mv12 ⇒ v1 = 
(D) Distance y-mass mp from y-axis, ri =
KE =
mω2(r12 + r22 + r32) = 
mω2 4a(y1 + y2 + y3)
(A) I =
m(x12 + z12) (distance of mi from y-axis is ) = 4ma (y1 + y2 + y3).(B) mg (x1 + x2 + x3) = mg(y1 + y2 + y3)
(C) mg x1 =
mv12 ⇒ v1 = (D) Distance y-mass mp from y-axis, ri =
KE =
mω2(r12 + r22 + r32) = mω2 4a(y1 + y2 + y3)Create a free account to view solution
View Solution FreeMore Rotational Motion Questions
A uniform disc of radius R lies in the x-y plane with its centre at the origin. Its moment of inertia about z axis is eq...A body is rolls on a rough curved surface with an initial speed v as shown in the figure if the body undergoing pure rol...A small block of mass ′m′ is rigidly attached at ′P′ to a ring of mass ′3m′and radiu...A rod hinged at one end is released from the horizontal position as shown in the figure. When it becomes vertical its lo...A sphere is placed rotating with its centre initially at rest i a corner as shown in figure (A) & (B). Coefficient of fr...