A particle of mass 0.6 kg moves in a horizontal circle with constant angular speed 1.5 radians per second. If the force directed towards the centre of the circle has magnitude 0.27 N , find the radius of the circular path.
The diagram shows a particle of mass 0.7 kg resting on a rough horizontal table. The coefficient of friction between the particle and the table is 0.25 . A light elastic string, of natural length 50 cm and modulus of elasticity 6.86 N , is attached to the particle. The string is kept at an angle of \(60 ^ { \circ }\) to the horizontal and is gradually extended by pulling on it until the particle moves. Show that the particle starts to move when the extension in the string is 17 cm .
A smooth circular hoop of radius 1 m , with centre \(O\), is fixed in a vertical plane. A small ring \(Q\), of mass 0.1 kg , is threaded onto the hoop and held so that the angle \(Q O H = 30 ^ { \circ }\), where \(H\) is the highest point of the hoop. \(Q\) is released from rest at this position. Find, in terms of \(g\),
the horizontal and vertical components of the acceleration of \(Q\) when it reaches the lowest
A particle \(P\) moves with simple harmonic motion in a straight line, with the centre of motion at the point \(O\) on the line. \(A\) and \(B\) are on opposite sides of \(O\), with \(O A = 4 \mathrm {~m} , O B = 6 \mathrm {~m}\).
When passing through \(A\) and \(B , P\) has speed \(6 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively.
Find the amplitude of the motion.
Show that the period of motion is \(2 \pi \mathrm {~s}\).
(a) Prove that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac { 3 r } { 8 }\) from its plane face.
A solid cylinder of radius \(\frac { 3 r } { 4 }\) and height \(k r\), where \(k < 1\), is welded to a uniform hemisphere of radius \(r\) made of the same material, so that their axes of symmetry coincide. The figure shows the cross section of the resulting solid. If the centre of mass of this solid is at \(O\), the centre of the plane face of the hemisphere,
\includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-1_191_172_2298_1805}