3.03r Friction: concept and vector form

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Edexcel Paper 3 2018 June Q7
8 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-20_264_698_246_685} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A wooden crate of mass 20 kg is pulled in a straight line along a rough horizontal floor using a handle attached to the crate.
The handle is inclined at an angle \(\alpha\) to the floor, as shown in Figure 1, where \(\tan \alpha = \frac { 3 } { 4 }\) The tension in the handle is 40 N .
The coefficient of friction between the crate and the floor is 0.14
The crate is modelled as a particle and the handle is modelled as a light rod.
Using the model,
  1. find the acceleration of the crate. The crate is now pushed along the same floor using the handle. The handle is again inclined at the same angle \(\alpha\) to the floor, and the thrust in the handle is 40 N as shown in Figure 2 below. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-20_220_923_1457_571} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
  2. Explain briefly why the acceleration of the crate would now be less than the acceleration of the crate found in part (a).
Edexcel Paper 3 Specimen Q7
8 marks Standard +0.3
  1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\).
A particle of mass \(m\) is placed on the plane and then projected up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is \(\mu\).
The particle moves up the plane with a constant deceleration of \(\frac { 4 } { 5 } \mathrm {~g}\).
  1. Find the value of \(\mu\). The particle comes to rest at the point \(A\) on the plane.
  2. Determine whether the particle will remain at \(A\), carefully justifying your answer.
OCR FM1 AS 2021 June Q4
14 marks Standard +0.3
4
2.4
&
B1 for each of two correct statements about the models.
If commenting on the accuracy of (a), must emphasise that (a) is very inaccurate or at least quite inaccurate
Do not allow e.g.
- model (a) is not very effective
- Neither model is accurate
- (a) and (b) are not very accurate
Clear comparison between the accuracy of the two models (must emphasise that (b) is fairly accurate or considerably more accurate than (a)), or other suitable distinct second comment
Do not allow e.g.
- model (b) is more accurate than model (a)
- (b) is not accurate
Do not allow statement claiming that resistance is proportional to speed, or to speed \({ } ^ { 2 }\)
Suitable comments for (a):
- is very inaccurate
- predicted speed is nearly three times the actual value
- constant resistance is not a suitable model
- both models underestimate the resistance (as top speed is lower than expected)
For the linear model (b)
- is fairly accurate (but probably underestimates the resistance at higher speeds)
- resistance is not proportional to speed but is a much better model than constant resistance
3(a)\(T _ { 2 } \cos \theta = m _ { 2 } g\) \(T _ { 2 } = \frac { m _ { 2 } \times 9.8 } { 0.8 } = 12.25 m _ { 2 }\)
M1
A1
[2]
1.1a
1.1
Resolving \(T _ { 2 }\) vertically and balancing forces on \(R\)
Do not allow extra forces present
Allow use of g, e.g. \(\frac { 5 } { 4 } g m _ { 2 }\)
In this solution \(\theta\) is the angle between \(R P\) and \(R A\) Sin may be seen instead if \(\theta\) is measured horizontally.
Do not allow incomplete expressions e.g. \(\frac { m _ { 2 } g } { \sin 53.13 }\)
3(b)(i)\(\begin{aligned}T _ { 2 } \cos \theta + m _ { 1 } g = T _ { 1 } \cos \theta
T _ { 1 } = T _ { 2 } + \frac { 9.8 m _ { 1 } } { 0.8 } =
\qquad 12.25 m _ { 2 } + 12.25 m _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right) \end{aligned}\)
M1
A1
[2]
3.1b
2.1
Vertical forces on \(P\); 3 terms including resolving of \(T _ { 1 }\); allow sign error
AG Dividing by \(\cos \theta ( = 0.8 )\), substituting their \(T _ { 2 }\) and rearranging
Allow 12.25 instead of \(\frac { 49 } { 4 }\)
Or \(T _ { 1 } \cos \theta = m _ { 1 } g + m _ { 2 } g\) (equation for the system as a whole)
At least one intermediate step must be seen
3(b)(ii)\(\begin{aligned}T _ { 1 } \sin \theta + T _ { 2 } \sin \theta = m _ { 1 } a
12.25 \left( m _ { 1 } + m _ { 2 } \right) \times 0.6 + 12.25 m _ { 2 } \times 0.6 = m _ { 1 } \times 0.6 \omega ^ { 2 }
\omega ^ { 2 } = \frac { 7.35 m _ { 1 } + 14.7 m _ { 2 } } { 0.6 m _ { 1 } } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } } \end{aligned}\)
M1
M1
A1
[3]
3.1b
1.1
2.1
NII horizontally for \(P ; 3\) terms including resolving of tensions; allow sign error
Substituting for \(T _ { 1 }\), their \(T _ { 2 } , \sin \theta\) and \(\alpha\)
AG Must see an intermediate step
Could see \(a\) or \(0.6 \omega ^ { 2 }\) or \(\frac { v ^ { 2 } } { 0.6 }\) or \(\omega ^ { 2 } r\) or \(\frac { v ^ { 2 } } { r } \sin \theta = 0.6\)
must be \(a = 0.6 \omega ^ { 2 }\)
3(c)\(\begin{aligned}\text { E.g } m _ { 1 } \gg m _ { 2 } \Rightarrow \frac { 2 m _ { 2 } } { m _ { 1 } } \approx 0 \text { or } \frac { 49 m _ { 2 } } { 4 m _ { 1 } } \approx 0
\omega \approx \sqrt { \frac { 49 m } { 4 m } } = 3.5 \end{aligned}\)
M1 A1
[2]
1.1
1.1
Allow argument such as if \(m _ { 1 } \gg m _ { 2 }\) then \(m _ { 1 } + 2 m _ { 2 } \approx m _ { 1 }\)
AG \(m\) may be missing
SC1 for result following argument that \(m _ { 2 }\) is negligible (by comparison with \(m _ { 1 }\) ) without justification, or using trial values of \(m _ { 1 }\) and \(m _ { 2 }\) with \(m _ { 1 } \gg m _ { 2 }\).
Do not allow the assumption that \(m _ { 2 } = 0\)
If using trial values, \(m _ { 1 }\) must be at least \(70 \times m _ { 2 }\) to give \(\omega = 3.5\) to 1 dp .
3\multirow{3}{*}{(d)}
\(v = r \omega = 0.6 \sqrt { \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 } }\)
Final energy \(= 2.5 \times g \times 1\) \(\text { Initial } \mathrm { KE } = \frac { 1 } { 2 } \times 2.5 \times 0.6 ^ { 2 } \times \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 }\)
Initial PE \(= 2.5 \times g \times 1.2 + 2.8 \times g \times 0.4\)
Energy loss \(= 17.8605 + 40.376 - 24.5 = 33.7365\)
M1
B1
M1
M1
A1
1.2
1.1
1.1
1.1
3.2a
Use of \(v = r \omega\) with values for \(m _ { 1 }\) and \(m _ { 2 }\)
(Assuming zero PE level at 2 m below \(A\); other values possible)
Do not allow use of \(\omega = 3.5\)
oe with different zero PE level awrt 33.7
( \(v = 3.78 , v ^ { 2 } = 14.2884\) )
NB \(\omega = 6.3\) (24.5)
(17.8605)
(40.376)
Alternate method \(v = r \omega = 0.6 \sqrt { \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 } }\)
Initial KE \(= \frac { 1 } { 2 } \times 2.5 \times 0.6 ^ { 2 } \times \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 }\)
\(\triangle P E\) for \(m _ { 1 } = \pm 2.5 \times 9.8 \times ( 0.8 - 1 )\)
\(\triangle P E\) for \(m _ { 2 } = \pm 2.8 \times 9.8 ( 1.6 - 2 )\)
Energy loss \(= 17.8605 + 4.9 + 10.976\)
M1
M1
M1
M1
A1
Use of \(v = r \omega\) with values for \(m _ { 1 }\) and \(m _ { 2 }\)
Or \(- \triangle P E\) \(= 2.5 \times 9.8 \times 0.2 + 2.8 \times 9.8 \times 0.4\)
awrt 33.7
( \(v = 3.78 , v ^ { 2 } = 14.2884\) ) \(\mathrm { NB } \omega = 6.3\)
(17.8605)
\(( \pm 4.9 )\)
\(( \pm 10.976 )\)
\(( \pm 15.876 )\)
Or 15.876 + 17.8605
[5]
Pre-U Pre-U 9794/3 2013 June Q9
9 marks Standard +0.3
9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}
  1. Copy and complete the diagram to show all the forces acting on the particle.
  2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
  3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.
Pre-U Pre-U 9795/2 2020 Specimen Q9
6 marks Standard +0.3
9 The diagram shows a uniform rod \(A B\) of length 40 cm and mass 2 kg placed with the end \(A\) resting against a smooth vertical wall and the end \(B\) on rough horizontal ground. The angle between \(A B\) and the horizontal is \(60 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{f4acd242-eb78-4124-bfa2-fdecaa188690-5_657_659_392_705} Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.
CAIE FP2 2009 November Q4
11 marks Challenging +1.8
A uniform rod \(AB\), of length \(2a\) and mass \(2m\), can rotate freely in a vertical plane about a smooth horizontal axis through \(A\). A small rough ring of mass \(m\) is threaded on the rod. The rod is held in a horizontal position with the ring at rest at the mid-point of the rod. The rod is released from rest. Using energy considerations, show that, until the ring slides, $$a\dot{\theta}^2 = \frac{18}{11}g \sin \theta,$$ where \(\theta\) is the angle turned through by the rod. [3] Show that, until the ring slides, the magnitudes of the friction force and normal contact force acting on the ring are \(\frac{20}{11}mg \sin \theta\) and \(\frac{2}{11}mg \cos \theta\) respectively. [6] The coefficient of friction between the ring and the rod is \(\mu\). Find, in terms of \(\mu\), the value of \(\theta\) when the ring starts to slide. [2]
CAIE FP2 2018 November Q4
11 marks Challenging +1.2
\includegraphics{figure_4} A uniform rod \(AB\) of length \(2a\) and weight \(W\) rests against a smooth horizontal peg at a point \(C\) on the rod, where \(AC = x\). The lower end \(A\) of the rod rests on rough horizontal ground. The rod is in equilibrium inclined at an angle of \(45°\) to the horizontal (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). The rod is about to slip at \(A\). \begin{enumerate}[label=(\roman*)] \item Find an expression for \(x\) in terms of \(a\) and \(\mu\). [5] \item Hence show that \(\mu \geqslant \frac{1}{3}\). [2] \item Given that \(x = \frac{5}{3}a\), find the value of \(\mu\) and the magnitude of the resultant force on the rod at \(A\). [4] \end{enumerate]
CAIE FP2 2019 November Q2
8 marks Challenging +1.2
\includegraphics{figure_2} A uniform square lamina \(ABCD\) of side \(4a\) and weight \(W\) rests in a vertical plane with the edge \(AB\) inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{1}{4}\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(AB\), where \(BE = 3a\) (see diagram).
  1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\). [5]
  2. Given that the lamina is about to slip, find the value of \(\mu\). [3]
CAIE M1 2020 June Q4
7 marks Standard +0.3
The diagram shows a ring of mass \(0.1\text{ kg}\) threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is \(0.8\). A force of magnitude \(T\text{ N}\) acts on the ring in a direction at \(30°\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest. \includegraphics{figure_4}
  1. Find the greatest value of \(T\) for which the ring remains at rest. [4]
  2. Find the acceleration of the ring when \(T = 3\). [3]
CAIE M1 2020 June Q7
10 marks Standard +0.3
A particle \(P\) of mass \(0.3\text{ kg}\), lying on a smooth plane inclined at \(30°\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of \(2.5\text{ m}\) and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass \(0.2\text{ kg}\) lies at rest on the horizontal plane \(1.5\text{ m}\) from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\). \includegraphics{figure_7}
  1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2\text{ m s}^{-1}\). Find the speed of \(Q\) after the collision. [5]
  2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2\text{ m s}^{-1}\). Find the coefficient of friction between \(P\) and the horizontal plane. [5]
CAIE M1 2020 June Q3
8 marks Standard +0.8
\includegraphics{figure_3} A particle of mass 2.5 kg is held in equilibrium on a rough plane inclined at 20° to the horizontal by a force of magnitude \(T\) N making an angle of 60° with a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.3. Find the greatest and least possible values of \(T\). [8]
CAIE M1 2021 June Q7
11 marks Standard +0.3
\includegraphics{figure_7} A slide in a playground descends at a constant angle of 30° for 2.5 m. It then has a horizontal section in the same vertical plane as the sloping section. A child of mass 35 kg, modelled as a particle \(P\), starts from rest at the top of the slide and slides straight down the sloping section. She then continues along the horizontal section until she comes to rest (see diagram). There is no instantaneous change in speed when the child goes from the sloping section to the horizontal section. The child experiences a resistance force on the horizontal section of the slide, and the work done against the resistance force on the horizontal section of the slide is 250 J per metre.
  1. It is given that the sloping section of the slide is smooth.
    1. Find the speed of the child when she reaches the bottom of the sloping section. [3]
    2. Find the distance that the child travels along the horizontal section of the slide before she comes to rest. [2]
  2. It is given instead that the sloping section of the slide is rough and that the child comes to rest on the slide 1.05 m after she reaches the horizontal section. Find the coefficient of friction between the child and the sloping section of the slide. [6]
CAIE M1 2022 June Q3
5 marks Standard +0.3
A crate of mass 300 kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5. A force of magnitude \(X\) N, acting at an angle \(\alpha\) above the horizontal, is applied to the crate, where \(\sin \alpha = 0.28\). Find the greatest value of \(X\) for which the crate remains at rest. [5]
CAIE M1 2014 June Q2
5 marks Easy -1.2
A block of mass \(2\) kg is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is \(0.3\).
  1. Calculate the maximum frictional force that can act on the block. [2]
  2. A horizontal force of \(5\) N is applied to the block. Calculate the acceleration of the block. [3]
CAIE M1 2018 June Q7
12 marks Standard +0.3
\includegraphics{figure_7} The diagram shows a triangular block with sloping faces inclined to the horizontal at \(45°\) and \(30°\). Particle \(A\) of mass \(0.8 \text{ kg}\) lies on the face inclined at \(45°\) and particle \(B\) of mass \(1.2 \text{ kg}\) lies on the face inclined at \(30°\). The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the faces. The parts \(AP\) and \(BP\) of the string are parallel to lines of greatest slope of the respective faces. The particles are released from rest with both parts of the string taut. In the subsequent motion neither particle reaches the pulley and neither particle reaches the bottom of a face.
  1. Given that both faces are smooth, find the speed of \(A\) after each particle has travelled a distance of \(0.4 \text{ m}\). [6]
  2. It is given instead that both faces are rough. The coefficient of friction between each particle and a face of the block is \(\mu\). Find the value of \(\mu\) for which the system is in limiting equilibrium. [6]
CAIE M1 2018 June Q5
6 marks Standard +0.3
A particle of mass \(3\text{ kg}\) is on a rough plane inclined at an angle of \(20°\) to the horizontal. A force of magnitude \(P\text{ N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is \(0.35\). Show that the least possible value of \(P\) is \(0.394\), correct to 3 significant figures, and find the greatest possible value of \(P\). [6]
CAIE M1 2019 June Q3
5 marks Moderate -0.3
A particle of mass 13 kg is on a rough plane inclined at an angle of \(\theta\) to the horizontal, where \(\tan \theta = \frac{5}{12}\). The coefficient of friction between the particle and the plane is 0.3. A force of magnitude \(T\) N, acting parallel to a line of greatest slope, moves the particle a distance of 2.5 m up the plane at a constant speed. Find the work done by this force. [5]
CAIE M1 2017 March Q3
6 marks Standard +0.3
\includegraphics{figure_3} A particle of mass \(0.6\) kg is placed on a rough plane which is inclined at an angle of \(21°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting parallel to a line of greatest slope of the plane, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.3\). Show that the least possible value of \(P\) is \(0.470\), correct to \(3\) significant figures, and find the greatest possible value of \(P\). [6]
CAIE M1 2019 March Q1
4 marks Moderate -0.3
\includegraphics{figure_1} A small ring \(P\) of mass \(0.03\) kg is threaded on a rough vertical rod. A light inextensible string is attached to the ring and is pulled upwards at an angle of \(15°\) to the horizontal. The tension in the string is \(2.5\) N (see diagram). The ring is in limiting equilibrium and on the point of sliding up the rod. Find the coefficient of friction between the ring and the rod. [4]
CAIE M1 2019 November Q3
7 marks Moderate -0.3
A block of mass 3 kg is at rest on a rough plane inclined at 60° to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of the plane is just sufficient to prevent the block from sliding down the plane.
  1. Find the coefficient of friction between the block and the plane. [5]
The force of magnitude 15 N is now replaced by a force of magnitude \(X\) N acting up the line of greatest slope.
  1. Find the greatest value of \(X\) for which the block does not move. [2]
CAIE M1 2019 November Q6
11 marks Standard +0.3
A block of mass 3 kg is initially at rest on a rough horizontal plane. A force of magnitude 6 N is applied to the block at an angle of \(\theta\) above the horizontal, where \(\cos \theta = \frac{24}{25}\). The force is applied for a period of 5 s, during which time the block moves a distance of 4.5 m.
  1. Find the magnitude of the frictional force on the block. [4]
  2. Show that the coefficient of friction between the block and the plane is 0.165, correct to 3 significant figures. [3]
  3. When the block has moved a distance of 4.5 m, the force of magnitude 6 N is removed and the block then decelerates to rest. Find the total time for which the block is in motion. [4]
CAIE M1 Specimen Q4
6 marks Standard +0.3
\includegraphics{figure_4} Blocks \(P\) and \(Q\), of mass \(m\) kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at 35° to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2. Find the set of values of \(m\) for which the two blocks remain at rest. [6]
CAIE M2 2010 June Q2
5 marks Standard +0.3
\includegraphics{figure_2} A uniform solid cone has height 30 cm and base radius \(r\) cm. The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(35°\), when the cone topples. The diagram shows a cross-section of the cone.
  1. Find the value of \(r\). [3]
  2. Show that the coefficient of friction between the cone and the plane is greater than 0.7. [2]
CAIE M2 2017 June Q7
9 marks Standard +0.8
A particle \(P\) of mass \(0.5\) kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t\) s the magnitude of the force is \(0.6t^2\) N and the velocity of \(P\) away from \(O\) is \(v\,\text{m}\,\text{s}^{-1}\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).
  1. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac{\text{d}v}{\text{d}t} = 1.2t^2 - 0.3 \quad \text{for } t > 0.5.$$ [3]
  2. Express \(v\) in terms of \(t\) for \(t > 0.5\). [3]
  3. Find the displacement of \(P\) from \(O\) when \(t = 1.2\). [3]
CAIE M2 2017 June Q7
9 marks Standard +0.3
A particle \(P\) of mass \(0.5\) kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t\) s the magnitude of the force is \(0.6t^2\) N and the velocity of \(P\) away from \(O\) is \(v \text{ ms}^{-1}\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).
  1. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac{\text{dv}}{\text{dt}} = 1.2t^2 - 0.3 \quad \text{for } t > 0.5.$$ [3]
  2. Express \(v\) in terms of \(t\) for \(t > 0.5\). [3]
  3. Find the displacement of \(P\) from \(O\) when \(t = 1.2\). [3]