String through hole/bead on string

3
\includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-3_637_572_264_788} One end of a light inextensible string is attached to a point \(C\). The other end is attached to a point \(D\), which is 1.1 m vertically below \(C\). A small smooth ring \(R\), of mass 0.2 kg , is threaded on the string and moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, with centre at \(O\) and radius 1.2 m , where \(O\) is 0.5 m vertically below \(D\) (see diagram).

1. Show that the tension in the string is 1.69 N , correct to 3 significant figures.
2. Find the value of \(v\).

6
\includegraphics[max width=\textwidth, alt={}, center]{e5d70ccb-cec0-4390-a500-b550957a4ac6-4_503_805_260_671} A light inextensible string passes through a small smooth bead \(B\) of mass 0.4 kg . One end of the string is attached to a fixed point \(A 0.4 \mathrm {~m}\) above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and 0.3 m above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius 0.3 m (see diagram).

1. Given that the tension in the string is 2 N , calculate
   (a) the angular speed of the bead,
   (b) the magnitude of the contact force exerted on the bead by the surface.
2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead.

6
\includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-4_503_805_260_671} A light inextensible string passes through a small smooth bead \(B\) of mass 0.4 kg . One end of the string is attached to a fixed point \(A 0.4 \mathrm {~m}\) above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and 0.3 m above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius 0.3 m (see diagram).

1. Given that the tension in the string is 2 N , calculate
   (a) the angular speed of the bead,
   (b) the magnitude of the contact force exerted on the bead by the surface.
2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead.

6
\includegraphics[max width=\textwidth, alt={}, center]{c403a227-586d-4c1f-a392-e475234fc0a0-10_262_732_264_705} A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a particle \(Q\) of mass 0.3 kg . The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length 0.3 m and modulus of elasticity 15 N joins \(Q\) to a fixed point \(A\) which is 0.4 m vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).

1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended.
2. Find the distance \(H P\) given that the angular speed of \(P\) is \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-10_262_732_264_705} A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a particle \(Q\) of mass 0.3 kg . The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length 0.3 m and modulus of elasticity 15 N joins \(Q\) to a fixed point \(A\) which is 0.4 m vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).

1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended.
2. Find the distance \(H P\) given that the angular speed of \(P\) is \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

6 A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m . One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is \(T \mathrm {~N}\), and the bead rotates with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle about the vertical axis through \(A\) and \(C\).

1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m , calculate \(\omega\), and hence find the kinetic energy of \(B\).
2. Given instead that angle \(A B C = 90 ^ { \circ }\), and that \(A B\) makes an angle \(\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)\) with the vertical, calculate \(T\) and \(\omega\).