Questions — CAIE M2 (519 questions)

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CAIE M2 2012 November Q4
6 marks Standard +0.3
\includegraphics{figure_4} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45°\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius \(0.67\) m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\). [6]
CAIE M2 2012 November Q5
7 marks Standard +0.3
A particle \(P\) is projected with speed \(30\) m s\(^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\) on horizontal ground. For the instant when the speed of \(P\) is \(17\) m s\(^{-1}\) and increasing,
  1. show that the vertical component of the velocity of \(P\) is \(8\) m s\(^{-1}\) downwards, [2]
  2. calculate the distance of \(P\) from \(O\). [5]
CAIE M2 2012 November Q6
8 marks Standard +0.3
\includegraphics{figure_6} A uniform lamina \(OABCD\) consists of a semicircle \(BCD\) with centre \(O\) and radius \(0.6\) m and an isosceles triangle \(OAB\), joined along \(OB\) (see diagram). The triangle has area \(0.36\) m\(^2\) and \(AB = AO\).
  1. Show that the centre of mass of the lamina lies on \(OB\). [4]
  2. Calculate the distance of the centre of mass of the lamina from \(O\). [4]
CAIE M2 2012 November Q7
12 marks Challenging +1.2
A light elastic string has natural length \(3\) m and modulus of elasticity \(45\) N. A particle \(P\) of weight \(6\) N is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\) and \(AB = 4\) m. The particle \(P\) is released from rest at the point \(1.5\) m vertically below \(A\).
  1. Calculate the distance \(P\) moves after its release before first coming to instantaneous rest at a point vertically above \(B\). (You may assume that at this point the part of the string joining \(P\) to \(B\) is slack.) [4]
  2. Show that the greatest speed of \(P\) occurs when it is \(2.1\) m below \(A\), and calculate this greatest speed. [5]
  3. Calculate the greatest magnitude of the acceleration of \(P\). [3]
CAIE M2 2013 November Q1
Standard +0.3
\includegraphics{figure_1}
  1. \(A\) has velocity \(\vec{x}\) and \(C\) has velocity \(\vec{v}\)
CAIE M2 2013 November Q2
Moderate -0.5
\includegraphics{figure_2}
    1. \(C\)
    2. \(C\) has velocity \(\vec{v}\)
CAIE M2 2013 November Q3
Moderate -1.0
\(A\) has velocity \(\vec{v}\), there are velocities \(\vec{x}\), \(\vec{v}\), \(\vec{v}\) around point \(O\), and velocity \(\vec{v}\)
    1. —
    2. \(v\) and \(v\)
CAIE M2 2013 November Q4
Moderate -0.3
\(A\) has velocity \(\vec{v}\)
    1. \(C\) with velocity \(\vec{v}\)
    2. \(C\)
CAIE M2 2013 November Q5
Moderate -0.5
\includegraphics{figure_5} \(A\) has velocity \(\vec{x}\), there are velocities \(\vec{x}\), \(\vec{x}\), \(\vec{v}\) and \(\vec{v}\)
    1. \(v\)
    2. \(v\) and \(v\)
CAIE M2 2013 November Q6
Moderate -0.5
\includegraphics{figure_6} \(E\) has velocity \(\vec{v}\)
    1. \(B\)
    2. \(v\)
CAIE M2 2013 November Q7
Moderate -0.5
\(A\) has velocity \(\vec{x}\)
    1. \(C\)
  2. \(A\) has velocity \(\vec{x}\)
    1. \(C\)
  3. \(C\) with velocities \(v \vec{v}\)
CAIE M2 2013 November Q1
8 marks Moderate -0.8
A particle moves in a straight line. Its displacement from a fixed point \(O\) at time \(t\) seconds is \(s\) metres, where \(s = t^3 - 9t^2 + 24t\).
  1. Find expressions for the velocity \(v\) and acceleration \(a\) of the particle at time \(t\).
  2. Find the values of \(t\) for which the particle is at rest.
  3. Find the total distance travelled by the particle in the first \(6\) seconds.
[8]
CAIE M2 2013 November Q2
6 marks Moderate -0.5
A particle moves in a straight line. At time \(t\) seconds its velocity is \(v\) ms\(^{-1}\) and its acceleration is \(a\) ms\(^{-2}\).
  1. Given that \(a = —\), express \(v\) in terms of \(t\).
  2. Given that \(v = tv\) when \(t = 0\), find \(v\) in terms of \(t\).
  3. Find the displacement from the starting point when \(t = v\).
[6]
CAIE M2 2013 November Q3
8 marks Standard +0.3
\includegraphics{figure_3} A particle moves on the inner surface of a smooth hollow cone of semi-vertical angle \(\alpha\). The axis of the cone is vertical with the vertex at the bottom. The particle moves in a horizontal circle of radius \(r\) with constant speed \(v\). Find expressions for the normal reactions on the particle from the cone surface, and show that the height of the particle above the vertex is \(\frac{v^2}{g \tan \alpha}\). [8]
CAIE M2 2013 November Q4
14 marks Standard +0.8
A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point. The particle moves in a vertical circle.
  1. Show that the speed \(v\) at the lowest point of the circle must satisfy \(v^2 \geq 5gl\) for the particle to complete the circle.
  2. Given that the particle just completes the circle, find the tensions in the string at the highest and lowest points of the circle.
  3. Given that \(v^2 = 6gl\) at the lowest point, find the tension in the string when the particle has risen through an angle \(\theta\) from the lowest point.
[14]
CAIE M2 2013 November Q5
8 marks Standard +0.3
A smooth sphere of mass \(M\) and radius \(a\) rests in contact with a smooth vertical wall and a smooth inclined plane. The plane makes an angle \(\alpha\) with the horizontal.
  1. Find the magnitude of each of the contact forces acting on the sphere.
  2. Find the range of values of \(\alpha\) for which this equilibrium is possible.
[8]
CAIE M2 2013 November Q6
8 marks Moderate -0.3
Two particles \(A\) and \(B\) have masses \(3m\) and \(2m\) respectively. Initially \(A\) is at rest and \(B\) is moving with speed \(u\) in a straight line towards \(A\). The coefficient of restitution between the particles is \(e\).
  1. Find the speeds of the particles immediately after the collision.
  2. Find the condition on \(e\) for \(A\) to be moving faster than \(B\) after the collision.
[8]
CAIE M2 2013 November Q7
16 marks Challenging +1.8
\includegraphics{figure_7} A uniform solid hemisphere of mass \(M\) and radius \(a\) is placed with its curved surface on rough horizontal ground. A horizontal force \(P\) is applied to the hemisphere at the centre of its flat circular face.
  1. Find the minimum value of the coefficient of friction \(\mu\) between the hemisphere and the ground for the hemisphere to slide without toppling.
  2. Show that if \(\mu < \frac{3}{8}\), the hemisphere will topple.
  3. Find the maximum horizontal distance that the centre of mass of the hemisphere moves before toppling begins, given that \(\mu = \frac{1}{4}\) and the hemisphere starts from rest.
  4. Find the angular acceleration of the hemisphere about its point of contact with the ground at the instant when toppling begins.
[16]
CAIE M2 2014 November Q1
7 marks Standard +0.8
A particle of mass \(m\) moves in a straight line. At time \(t\), its displacement from a fixed point on the line is \(s\) and its velocity is \(v\). The particle experiences a retarding force of magnitude \(mkv^2\), where \(k\) is a positive constant. Find the relationship between \(v\) and \(t\). [7]
CAIE M2 2014 November Q2
6 marks Standard +0.3
\includegraphics{figure_2} A uniform rod \(AB\) of mass \(3m\) and length \(4a\) rests in equilibrium in a vertical plane with the end \(A\) on rough horizontal ground and the end \(B\) against a smooth vertical wall. The rod makes an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac{3}{5}\).
  1. Find the normal reaction at \(A\) and the normal reaction at \(B\). [4]
  2. Find the coefficient of friction between the rod and the ground. [2]
CAIE M2 2014 November Q3
6 marks Moderate -0.3
A particle \(P\) of mass \(0.2\) kg 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 moves in a horizontal circle of radius \(0.8\) m with the string making a constant angle of \(60°\) with the vertical. Calculate the speed of the particle and the tension in the string. [6]
CAIE M2 2014 November Q4
8 marks Standard +0.3
\includegraphics{figure_4} The diagram shows the cross-section of a uniform solid consisting of a cylinder of radius \(0.4\) m and height \(1.5\) m with a hemisphere of radius \(0.4\) m on top.
  1. Find the distance of the centre of mass above the base of the cylinder. [5]
  2. The solid can just rest in equilibrium on a plane inclined at angle \(\alpha\) to the horizontal. Find \(\alpha\). [3]
CAIE M2 2014 November Q5
7 marks Moderate -0.3
The position vector of a particle at time \(t\) is given by \(\mathbf{r} = t^2\mathbf{i} + (3t - 1)\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. Find the velocity and acceleration of the particle when \(t = 2\).
  1. Hence find the angle between the velocity and acceleration vectors when \(t = 2\). [3]
  2. Find the value of \(t\) for which the velocity and acceleration vectors are perpendicular. [4]
CAIE M2 2014 November Q6
12 marks Standard +0.3
A particle of mass \(2\) kg moves under the action of a variable force. At time \(t\) seconds the force is \((6t - 3)\mathbf{i} + 4\mathbf{j}\) newtons, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. When \(t = 0\), the particle is at rest at the origin.
  1. Find the velocity of the particle when \(t = 4\). [4]
  2. Find the kinetic energy of the particle when \(t = 4\). [2]
  3. Find the distance of the particle from the origin when \(t = 2\). [6]
CAIE M2 2014 November Q7
8 marks Standard +0.3
\includegraphics{figure_7} A particle of mass \(0.4\) kg is attached to one end of a light inextensible string of length \(2\) m. The other end of the string is attached to a fixed point \(O\). The particle moves in a vertical circle and passes through the lowest point of the circle with speed \(6\) m s\(^{-1}\).
  1. Find the tension in the string when the particle is at the lowest point. [2]
  2. Find the speed of the particle when the string makes an angle of \(60°\) with the downward vertical. [4]
  3. Hence find the tension in the string at this position. [2]