Conical pendulum (horizontal circle)

1
\includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-2_381_1079_255_534} A particle \(P\) of mass 0.4 kg is attached to a fixed point \(A\) by a light inextensible string. The string is inclined at \(60 ^ { \circ }\) to the vertical. \(P\) moves with constant speed in a horizontal circle of radius 0.2 m . The centre of the circle is vertically below \(A\) (see diagram).

1. Show that the tension in the string is 8 N .
2. Calculate the speed of the particle.

4 One end of a light inextensible string of length 2.4 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(0.2 \mathrm {~kg} . P\) moves with constant speed in a horizontal circle which has its centre vertically below \(A\), with the string taut and making an angle of \(60 ^ { \circ }\) with the vertical.

1. Find the speed of \(P\). The string of length 2.4 m is removed, and \(P\) is now connected to \(A\) by a light inextensible string of length 1.2 m . The particle \(P\) moves with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with its centre vertically below \(A\).
2. Calculate the angle between the string and the vertical.

3
\includegraphics[max width=\textwidth, alt={}, center]{7958ad9d-1a28-439d-8de0-5039f2dba770-2_255_926_1073_614} One end of a light inextensible string is attached to a fixed point \(A\) and the other end of the string is attached to a particle \(P\). The particle \(P\) moves with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle which has its centre \(O\) vertically below \(A\). The string makes an angle \(\theta\) with the vertical (see diagram). The tension in the string is three times the weight of \(P\).

1. Show that the length of the string is 1.2 m .
2. Find the speed of \(P\).
   \includegraphics[max width=\textwidth, alt={}, center]{7958ad9d-1a28-439d-8de0-5039f2dba770-3_521_1004_258_575} A small ball \(B\) is projected from a point \(O\) above horizontal ground, with initial speed \(15 \mathrm {~ms} ^ { - 1 }\) at an angle of projection of \(30 ^ { \circ }\) above the horizontal (see diagram). The ball strikes the ground 3 s after projection.
3. Calculate the speed and direction of motion of the ball immediately before it strikes the ground.
4. Find the height of \(O\) above the ground.

7 A particle \(P\) of mass 0.7 kg is attached to one end of a light inextensible string of length 0.5 m . The other end of the string is attached to a fixed point \(A\) which is \(h \mathrm {~m}\) above a smooth horizontal surface. \(P\) moves in contact with the surface with uniform circular motion about the point on the surface which is vertically below \(A\).

1. Given that \(h = 0.14\), find an inequality for the angular speed of \(P\).
2. Given instead that the magnitude of the force exerted by the surface on \(P\) is 1.4 N and that the speed of \(P\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate the tension in the string and the value of \(h\).

1
\includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-03_231_970_258_591} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.8 m . The fixed point \(O\) is 0.15 m vertically below \(A\). The particle \(P\) moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with centre \(O\) (see diagram).

1. Show that the tension in the string is 16 N .
2. Find the value of \(v\).

1
\includegraphics[max width=\textwidth, alt={}, center]{111bcbf6-daaf-4d8d-9299-d591ac7369f1-03_231_970_258_591} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.8 m . The fixed point \(O\) is 0.15 m vertically below \(A\). The particle \(P\) moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with centre \(O\) (see diagram).

1. Show that the tension in the string is 16 N .
2. Find the value of \(v\).

7
\includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-3_451_432_1434_852} One end of a light inextensible string is attached to the highest point \(A\) of a solid fixed sphere with centre \(O\) and radius 0.6 m . The other end of the string is attached to a particle \(P\) of mass 0.2 kg which rests in contact with the smooth surface of the sphere. The angle \(A O P = 60 ^ { \circ }\) (see diagram). The sphere exerts a contact force of magnitude \(R \mathrm {~N}\) on \(P\) and the tension in the string is \(T \mathrm {~N}\).

1. By resolving vertically, show that \(R + ( \sqrt { } 3 ) T = 4\).
   \(P\) is now set in motion, and moves with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on the surface of the sphere.
2. Find an equation involving \(R , T\) and \(\omega\).
3. Hence
   (a) calculate \(R\) when \(\omega = 2\),
   (b) find the greatest possible value of \(T\) and the corresponding speed of \(P\).

1 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) on a smooth horizontal plane. The particle \(P\) moves in horizontal circles about \(O\). The tension in the string is \(4 m g\). Find, in terms of \(a\) and \(g\), the time that \(P\) takes to make one complete revolution.

2 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) which is 0.4 m vertically below \(A\).

1. Show that the tension in the string is 2.5 N .
2. Find the speed of \(P\).

2 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) which is 0.4 m vertically below \(A\).

1. Show that the tension in the string is 2.5 N .
2. Find the speed of \(P\).

2

1. A particle P , of mass 48 kg , is moving in a horizontal circle of radius 8.4 m at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), in contact with a smooth horizontal surface. A light inextensible rope of length 30 m connects P to a fixed point A which is vertically above the centre C of the circle, as shown in Fig. 2.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-3_526_490_482_870} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure}
   1. Given that \(V = 3.5\), find the tension in the rope and the normal reaction of the surface on P .
   2. Calculate the value of \(V\) for which the normal reaction is zero.
2. The particle P , of mass 48 kg , is now placed on the highest point of a fixed solid sphere with centre O and radius 2.5 m . The surface of the sphere is smooth. The particle P is given an initial horizontal velocity of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and it then moves in part of a vertical circle with centre O and radius 2.5 m . When OP makes an angle \(\theta\) with the upward vertical and P is still in contact with the surface of the sphere, P has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction of the sphere on P is \(R \mathrm {~N}\), as shown in Fig. 2.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-3_590_617_1706_804} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
   1. Show that \(v ^ { 2 } = u ^ { 2 } + 49 - 49 \cos \theta\).
   2. Find an expression for \(R\) in terms of \(u\) and \(v\).
   3. Given that P loses contact with the surface of the sphere at the instant when its speed is \(4.15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(u\).

2 A particle P of mass 0.3 kg is connected to a fixed point O by a light inextensible string of length 4.2 m . Firstly, P is moving in a horizontal circle as a conical pendulum, with the string making a constant angle with the vertical. The tension in the string is 3.92 N .

1. Find the angle which the string makes with the vertical.
2. Find the speed of P . P now moves in part of a vertical circle with centre O and radius 4.2 m . When the string makes an angle \(\theta\) with the downward vertical, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2). You are given that \(v = 8.4\) when \(\theta = 60 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2a4afead-e772-4d86-bc8d-86ffa5bca507-2_382_648_1985_751} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
3. Find the tension in the string when \(\theta = 60 ^ { \circ }\).
4. Show that \(v ^ { 2 } = 29.4 + 82.32 \cos \theta\).
5. Find \(\theta\) at the instant when the string becomes slack.

3 A particle \(P\) of mass 3.5 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 75 N . The other end of the string is attached to a fixed point \(O\). The particle rotates in a horizontal circle with a constant angular velocity of \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). The centre of the circle is vertically below \(O\). The magnitude of the tension in the string is \(T \mathrm {~N}\) and the length of the extended string is \(L \mathrm {~m}\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{a8c9d007-e67f-4637-9e74-630ba9a91442-3_460_424_447_817}

1. By considering the acceleration of \(P\), show that \(T = 31.5 L\).
2. Write down another relationship between \(T\) and \(L\).
3. Find the value of \(T\) and the value of \(L\).
4. Find the angle that the string makes with the downwards vertical through \(O\).

3 A right circular cone \(C\) of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of \(C\). The other end of the string is attached to a particle \(P\) of mass 2.5 kg .
\(P\) moves in a horizontal circle with constant speed and in contact with the smooth curved surface of \(C\). The extension of the string is 1.5 m .

1. Find the tension in the string.
2. Find the speed of \(P\).