3.03n Equilibrium in 2D: particle under forces

5 \includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_593_828_1530_660} A horizontal disc of radius 0.5 m is rotating with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed vertical axis through its centre \(O\). One end of a light inextensible string of length 0.8 m is attached to a point \(A\) of the circumference of the disc. A particle \(P\) of mass 0.4 kg is attached to the other end of the string. The string is taut and the system rotates so that the string is always in the same vertical plane as the radius \(O A\) of the disc. The string makes a constant angle \(\theta\) with the vertical (see diagram). The speed of \(P\) is 1.6 times the speed of \(A\).

1. Show that \(\sin \theta = \frac { 3 } { 8 }\).
2. Find the tension in the string.
3. Find the value of \(\omega\).

6 \includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-4_447_736_269_701} \(P\) is the vertex of a uniform solid cone of mass 5 kg , and \(O\) is the centre of its base. Strings are attached to the cone at \(P\) and at \(O\). The cone hangs in equilibrium with \(P O\) horizontal and the strings taut. The strings attached at \(P\) and \(O\) make angles of \(\theta ^ { \circ }\) and \(20 ^ { \circ }\), respectively, with the vertical (see diagram, which shows a cross-section).

1. By taking moments about \(P\) for the cone, find the tension in the string attached at \(O\).
2. Find the value of \(\theta\) and the tension in the string attached at \(P\).

4 A particle \(P\) of mass \(m\) is moving in a horizontal circle with angular speed \(\omega\) on the smooth inner surface of a hemispherical shell of radius \(r\). The angle between the vertical and the normal reaction of the surface on \(P\) is \(\theta\).

1. Show that \(\cos \theta = \frac { \mathrm { g } } { \omega ^ { 2 } \mathrm { r } }\).
   The plane of the circular motion is at a height \(x\) above the lowest point of the shell. When the angular speed is doubled, the plane of the motion is at a height \(4 x\) above the lowest point of the shell.
2. Find \(x\) in terms of \(r\).

1 A particle \(P\) is projected with speed \(u\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The particle reaches its greatest height at time \(T\) after projection. Find, in terms of \(u\), the speed of \(P\) at time \(\frac { 2 } { 3 } T\) after projection. \includegraphics[max width=\textwidth, alt={}, center]{7251b13f-1fae-4138-80ea-e6b8091c94ab-04_362_750_258_653} A light inextensible string of length \(a\) is threaded through a fixed smooth ring \(R\). One end of the string is attached to a particle \(A\) of mass \(3 m\). The other end of the string is attached to a particle \(B\) of mass \(m\). The particle \(A\) hangs in equilibrium at a distance \(x\) vertically below the ring. The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram). The particle \(B\) moves in a horizontal circle with constant angular speed \(2 \sqrt { \frac { \mathrm {~g} } { \mathrm { a } } }\). Show that \(\cos \theta = \frac { 1 } { 3 }\) and find \(x\) in terms of \(a\).

6. \begin{figure}[h] \end{figure} A particle of weight 120 N is placed on a fixed rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\).
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).
The particle is held at rest in equilibrium by a horizontal force of magnitude 30 N , which acts in the vertical plane containing the line of greatest slope of the plane through the particle, as shown in Figure 2.

1. Show that the normal reaction between the particle and the plane has magnitude 114 N . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-10_433_774_1464_604} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The horizontal force is removed and replaced by a force of magnitude \(P\) newtons acting up the slope along the line of greatest slope of the plane through the particle, as shown in Figure 3. The particle remains in equilibrium.
2. Find the greatest possible value of \(P\).
3. Find the magnitude and direction of the frictional force acting on the particle when \(P = 30\).

3. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a particle \(P\) are given by $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 7 \mathbf { i } - 9 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 2 } = ( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N } \end{aligned}$$ where \(p\) and \(q\) are constants.
Given that \(P\) is in equilibrium,

1. find the value of \(p\) and the value of \(q\). The force \(\mathbf { F } _ { 3 }\) is now removed. The resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is \(\mathbf { R }\). Find
2. the magnitude of \(\mathbf { R }\),
3. the angle, to the nearest degree, that the direction of \(\mathbf { R }\) makes with \(\mathbf { j }\).

4. \begin{figure}[h] \end{figure} A small parcel of mass 3 kg is held in equilibrium on a rough plane by the action of a horizontal force of magnitude 30 N acting in a vertical plane through a line of greatest slope. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 3. The parcel is modelled as a particle. The parcel is on the point of moving up the slope.

1. Draw a diagram showing all the forces acting on the parcel.
2. Find the normal reaction on the parcel.
3. Find the coefficient of friction between the parcel and the plane.

1. \begin{figure}[h] \end{figure} A particle of weight \(W\) newtons is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(50 ^ { \circ }\) respectively, as shown in Figure 1. Given that the tension in \(B C\) is 6 N , find

1. the tension in \(A C\),
2. the value of \(W\).

3. \begin{figure}[h] \end{figure} A particle of mass 2 kg is suspended from a horizontal ceiling by two light inextensible strings, \(P R\) and \(Q R\). The particle hangs at \(R\) in equilibrium, with the strings in a vertical plane. The string \(P R\) is inclined at \(55 ^ { \circ }\) to the horizontal and the string \(Q R\) is inclined at \(35 ^ { \circ }\) to the horizontal, as shown in Figure 1. \section*{Find}

1. the tension in the string \(P R\),
2. the tension in the string \(Q R\).

1. \begin{figure}[h] \end{figure} A particle of weight \(W\) is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with the strings in a vertical plane and with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(45 ^ { \circ }\) respectively, as shown in Figure 1. Find, in terms of \(W\),

1. the tension in \(A C\),
2. the tension in \(B C\).
   VILLI SIHI NITIIIUM ION OC
   VILV SIHI NI JAHM ION OC
   VI4V SIHI NI JIIIM ION OC

5. \begin{figure}[h] \end{figure} A small metal box of mass 6 kg is attached at \(B\) to two ropes \(B P\) and \(B Q\). The fixed points \(P\) and \(Q\) are on a horizontal ceiling and \(P Q = 3.5 \mathrm {~m}\). The box hangs in equilibrium at a vertical distance of 2 m below the line \(P Q\), with the ropes in a vertical plane and with angle \(B Q P = 45 ^ { \circ }\), as shown in Figure 3. The box is modelled as a particle and the ropes are modelled as light inextensible strings. Find

1. the tension in \(B P\),
2. the tension in \(B Q\).

4. \begin{figure}[h] \end{figure} Two identical small rings, \(A\) and \(B\), each of mass \(m\), are threaded onto a rough horizontal wire. The rings are connected by a light inextensible string. A particle \(C\) of mass \(3 m\) is attached to the midpoint of the string. The particle \(C\) hangs in equilibrium below the wire with angle \(B A C = \beta\), as shown in Figure 2. The tension in each of the parts, \(A C\) and \(B C\), of the string is \(T\)

1. By considering particle \(C\), find \(T\) in terms of \(m , g\) and \(\beta\)
2. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction between the wire and \(A\). The coefficient of friction between each ring and the wire is \(\frac { 4 } { 5 }\) The two rings, \(A\) and \(B\), are on the point of sliding along the wire towards each other.
3. Find the value of \(\tan \beta\) \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-11_2255_50_314_34}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

3. \begin{figure}[h] \end{figure} A parcel of mass 20 kg is at rest on a rough horizontal floor. The coefficient of friction between the parcel and the floor is 0.3 Two forces, both acting in the same vertical plane, of magnitudes 200 N and \(T \mathrm {~N}\) are applied to the parcel. The line of action of the 200 N force makes an angle of \(15 ^ { \circ }\) with the horizontal and the line of action of the \(T \mathrm {~N}\) force makes an angle of \(25 ^ { \circ }\) with the horizontal, as shown in Figure 1. The parcel is modelled as a particle \(P\). Find the smallest value of \(T\) for which \(P\) remains in equilibrium.

1. \begin{figure}[h] \end{figure} A particle \(P\) of weight 5 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held in equilibrium by a force of magnitude \(F\) newtons. The direction of this force is perpendicular to the string and \(O P\) makes an angle of \(60 ^ { \circ }\) with the vertical, as shown in Figure 1. Find

1. the value of \(F\)
2. the tension in the string.

6. \begin{figure}[h] \end{figure} A boat is pulled along a river at a constant speed by two ropes.
The banks of the river are parallel and the boat travels horizontally in a straight line, parallel to the riverbanks.

• The tension in the first rope is 500 N acting at an angle of \(40 ^ { \circ }\) to the direction of motion, as shown in Figure 3.
• The tension in the second rope is \(P\) newtons, acting at an angle of \(\alpha ^ { \circ }\) to the direction of motion, also shown in Figure 3.
• The resistance to motion of the boat as it moves through the water is a constant force of magnitude 900 N

The boat is modelled as a particle. The ropes are modelled as being light and lying in a horizontal plane. Use the model to find

1. the value of \(\alpha\)
2. the value of \(P\)

1. \begin{figure}[h] \end{figure} Figure 1 shows a small smooth ring threaded onto a light inextensible string.
One end of the string is attached to a fixed point \(A\) on a horizontal ceiling and the other end of the string is attached to a fixed point \(B\) on the ceiling. A horizontal force of magnitude 2 N acts on the ring so that the ring rests in equilibrium at a point \(C\), vertically below \(B\), with the string taut. The line of action of the horizontal force and the string both lie in the same vertical plane. The angle that the string makes with the ceiling at \(A\) is \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\) The tension in the string is \(T\) newtons. The mass of the ring is \(M \mathrm {~kg}\).

1. Find the value of \(T\)
2. Find the value of \(M\)

8. \begin{figure}[h] \end{figure} A fixed rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) A small smooth pulley is fixed at the top of the plane.
One end of a light inextensible string is attached to a particle \(P\) which is at rest on the plane. The string passes over the pulley and the other end of the string is attached to a particle \(Q\) which hangs vertically below the pulley, as shown in Figure 5. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(0.5 m\) The string from \(P\) to the pulley lies along a line of greatest slope of the plane.
The coefficient of friction between \(P\) and the plane is \(\mu\).
The system is in limiting equilibrium with the string taut and \(P\) is on the point of slipping up the plane.

1. Find the value of \(\mu\). The string breaks and \(P\) begins to move down the plane.
   When particle \(P\) has travelled a distance of 0.8 m down the plane, the speed of \(P\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find the value of \(V\).

3. A particle \(P\) of mass 1.5 kg is placed at a point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.6

1. Show that \(P\) rests in equilibrium at \(A\). A horizontal force of magnitude \(X\) newtons is now applied to \(P\), as shown in Figure 1. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{edcc4603-f006-4c4f-a4e5-063cab41da98-04_236_584_667_680} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} The particle is on the point of moving up the plane.
2. Find
   1. the magnitude of the normal reaction of the plane on \(P\),
   2. the value of \(X\).

8. \begin{figure}[h] \end{figure} Two particles, \(P\) and \(Q\), with masses \(2 m\) and \(m\) respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the edge of a rough horizontal table. Particle \(Q\) is held at rest on the table and particle \(P\) is on the surface of a smooth inclined plane. The top of the plane coincides with the edge of the table. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 4. The string lies in a vertical plane containing the pulley and a line of greatest slope of the plane. The coefficient of friction between \(Q\) and the table is \(\frac { 1 } { 2 }\). Particle \(Q\) is released from rest with the string taut and \(P\) begins to slide down the plane.

1. By writing down an equation of motion for each particle,
   1. find the initial acceleration of the system,
   2. find the tension in the string. Suppose now that the coefficient of friction between \(Q\) and the table is \(\mu\) and when \(Q\) is released it remains at rest.
2. Find the smallest possible value of \(\mu\).

   Leave
   blank
   Q8

7. \begin{figure}[h] \end{figure} A washing line \(A B C D\) is fixed at the points \(A\) and \(D\). There are two heavy items of clothing hanging on the washing line, one fixed at \(B\) and the other fixed at \(C\). The washing line is modelled as a light inextensible string, the item at \(B\) is modelled as a particle of mass 3 kg and the item at \(C\) is modelled as a particle of mass \(M \mathrm {~kg}\). The section \(A B\) makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), the section \(B C\) is horizontal and the section \(C D\) makes an angle \(\beta\) with the horizontal, where \(\tan \beta = \frac { 12 } { 5 }\), as shown in Figure 2. The system is in equilibrium.

1. Find the tension in \(A B\).
2. Find the tension in BC.
3. Find the value of \(M\).
   

   END

3. \begin{figure}[h] \end{figure} A particle of mass 10 kg is placed on a fixed rough inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). The particle is held in equilibrium by a force of magnitude \(P\) newtons, which acts up the plane, as shown in Figure 1. The line of action of the force lies in a vertical plane that contains a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).

1. Find the normal reaction between the particle and the plane.
2. Find the greatest possible value of \(P\).
3. Find the least possible value of \(P\). DO NOT WRITEIN THIS AREA
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

3. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.] Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\), are given by $$\mathbf { F } _ { 1 } = ( 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 2 } = ( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 3 } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }$$ where \(a\) and \(b\) are constants.
The forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a particle \(P\) of mass 4 kg .
Given that \(P\) rests in equilibrium on a smooth horizontal surface under the action of these three forces,

1. find the size of the angle between the direction of \(\mathbf { F } _ { 3 }\) and the direction of \(- \mathbf { j }\). The force \(\mathbf { F } _ { 3 }\) is now removed and replaced by the force \(\mathbf { F } _ { 4 }\) given by \(\mathbf { F } _ { 4 } = \lambda ( \mathbf { i } + 3 \mathbf { j } )\) N, where \(\lambda\) is a positive constant. When the three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 4 }\) act on \(P\), the acceleration of \(P\) has magnitude \(3.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(\lambda\).
   

1. A particle \(P\) rests in equilibrium on a smooth horizontal plane.

A system of three forces, \(\mathbf { F } _ { 1 } \mathrm {~N} , \mathbf { F } _ { 2 } \mathrm {~N}\) and \(\mathbf { F } _ { 3 } \mathrm {~N}\) where $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 3 c \mathbf { i } + 4 c \mathbf { j } ) \\ & \mathbf { F } _ { 2 } = ( - 14 \mathbf { i } + 7 \mathbf { j } ) \end{aligned}$$ is applied to \(P\).
Given that \(P\) remains in equilibrium,

1. find \(\mathbf { F } _ { 3 }\) in terms of \(c\), \(\mathbf { i }\) and \(\mathbf { j }\). The force \(\mathbf { F } _ { 3 }\) is removed from the system.
   Given that \(c = 2\)
2. find the size of the angle between the direction of \(\mathbf { i }\) and the direction of the resultant force acting on \(P\). The mass of \(P\) is \(m \mathrm {~kg}\).
   Given that the magnitude of the acceleration of \(P\) is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
3. find the value of \(m\).

5. \begin{figure}[h] \end{figure} A small bead of mass 0.2 kg is attached to the end \(P\) of a light rod \(P Q\). The bead is threaded onto a fixed vertical rough wire. The bead is held in equilibrium with the \(\operatorname { rod } P Q\) inclined to the wire at an angle \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 2. The thrust in the rod is \(T\) newtons.
The bead is modelled as a particle.

1. Find the magnitude and direction of the friction force acting on the bead when \(T = 2.5\) The coefficient of friction between the bead and the wire is \(\mu\).
   Given that the greatest possible value of \(T\) is 6.125
2. find the value of \(\mu\).

5. \begin{figure}[h] \end{figure} A small ring of mass 0.2 kg is attached to one end of a light inextensible string.
The ring is threaded onto a fixed rough vertical rod.
The string is taut and makes an angle \(\theta\) with the rod, as shown in Figure 3, where \(\tan \theta = \frac { 12 } { 5 }\) Given that the ring is in equilibrium and that the tension in the string is 10 N ,

1. find the magnitude of the frictional force acting on the ring,
2. state the direction of the frictional force acting on the ring. The coefficient of friction between the ring and the rod is \(\frac { 1 } { 4 }\) Given that the ring is in equilibrium, and that the tension in the string, \(T\) newtons, can now vary,
3. 1. find the minimum value of \(T\)
   2. find the maximum value of \(T\)