| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2014 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Elastic string on rough inclined plane |
| Difficulty | Challenging +1.2 This is a multi-part M2 elastic strings question requiring resolution of forces, energy methods, and understanding of friction direction changes. While it involves several steps and careful bookkeeping of energy components (elastic PE, gravitational PE, work against friction), the techniques are standard for M2 and the question provides clear guidance through its structure. The 'show that' part scaffolds the solution, and the energy equation setup is routine for this topic, though the algebra requires care. |
| Spec | 3.03v Motion on rough surface: including inclined planes6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
TOTAL
Question 8:
8
TOTAL
Answer all questions.
Answer each question in the space provided for that question.
1 An eagle has caught a salmon of mass 3kg to take to its nest. When the eagle is
flying with speed 8ms(cid:2)1, it drops the salmon. The salmon falls a vertical distance of
13 metres back into the sea.
The salmon is to be modelled as a particle. The salmon’s weight is the only force that
acts on it as it falls to the sea.
(a) Calculate the kinetic energy of the salmon when it is dropped by the eagle.
[2 marks]
(b) Calculate the potential energy lost by the salmon as it falls to the sea.
[2 marks]
(c) (i) Find the kinetic energy of the salmon when it reaches the sea.
[2 marks]
(ii) Hence find the speed of the salmon when it reaches the sea.
[2 marks]
2 A particle has mass 6kg. A single force ð24e(cid:2)2ti(cid:2)12t3jÞ newtons acts on the
particle at time t seconds. No other forces act on the particle.
(a) Find the acceleration of the particle at time t.
[2 marks]
(b) At time t ¼ 0, the velocity of the particle is ð(cid:2)7i(cid:2)4jÞms(cid:2)1.
Find the velocity of the particle at time t.
[4 marks]
(c) Find the speed of the particle when t ¼ 0:5.
[4 marks]
3 Four tools are attached to a board.
The board is to be modelled as a uniform lamina and the four tools as four particles.
The diagram shows the lamina, the four particles A, B, C and D, and the x and y axes.
y
C
B
D
A
O x
The lamina has mass 5kg and its centre of mass is at the point ð7, 6Þ.
Particle A has mass 4kg and is at the point ð11, 2Þ.
Particle B has mass 3kg and is at the point ð3, 6Þ.
Particle C has mass 7kg and is at the point ð5, 9Þ.
Particle D has mass 1kg and is at the point ð1, 4Þ.
Find the coordinates of the centre of mass of the system of board and tools.
[5 marks]
4 A particle, of mass 0.8kg, is attached to one end of a light inextensible string. The
other end of the string is attached to the fixed point O. The particle is set in motion,
so that it moves in a horizontal circle at constant speed, with the string at an angle of
35(cid:2) to the vertical. The centre of this circle is vertically below O, as shown in the
diagram.
O
35(cid:2)
The particle moves in a horizontal circle and completes 20 revolutions each minute.
(a) Find the angular speed of the particle in radians per second.
[2 marks]
(b) Find the tension in the string.
[3 marks]
(c) Find the radius of the horizontal circle.
[4 marks]
5 A light inextensible string, of length a, has one end attached to a fixed point O. A
particle, of mass m, is attached to the other end of the string. The particle is moving
in a vertical circle with centre O. The point Q is the highest point of the particle’s
path. When the particle is at P, vertically below O, the string is taut and the particle is
p
moving with speed 7 ffiaffiffigffiffi , as shown in the diagram.
Q
O
a
p
7 ffiaffiffigffiffi
P
(a) Find, in terms of g and a, the speed of the particle at the point Q.
[4 marks]
(b) Find, in terms of g and m, the tension in the string when the particle is at Q.
[3 marks]
6 A puck, of mass mkg, is moving in a straight line across smooth horizontal ice. At
time t seconds, the puck has speed vms(cid:2)1. As the puck moves, it experiences an air
1
resistance force of magnitude 0:3mv3 newtons, until it comes to rest. No other
horizontal forces act on the puck.
When t ¼ 0, the speed of the puck is 8ms(cid:2)1.
Model the puck as a particle.
(a) Show that
3
v ¼ ð4(cid:2)0:2tÞ2
[6 marks]
(b) Find the value of t when the puck comes to rest.
[2 marks]
(c) Find the distance travelled by the puck as its speed decreases from 8ms(cid:2)1 to zero.
[5 marks]
7 A uniform ladder AB, of length 6 metres and mass 22kg, rests with its foot, A, on
rough horizontal ground. The ladder rests against the top of a smooth vertical wall at
the point C, where the length AC is 5 metres. The vertical plane containing the ladder
is perpendicular to the wall, and the angle between the ladder and the ground is 60(cid:2).
A man, of mass 88kg, is standing on the ladder.
The man may be modelled as a particle at the point D, where the length of AD is
4 metres.
The ladder is on the point of slipping.
B
1m
C
1m
D
4m
60(cid:2)
A
(a) Draw a diagram to show the forces acting on the ladder.
[2 marks]
(b) Find the coefficient of friction between the ladder and the horizontal ground.
[6 marks]An elastic string has natural length 1.5 metres and modulus of elasticity 120 newtons. One end of the string is attached to a fixed point, $A$, on a rough plane inclined at $20°$ to the horizontal. The other end of the elastic string is attached to a particle of mass 4 kg. The coefficient of friction between the particle and the plane is 0.8.
The three points, $A$, $B$ and $C$, lie on a line of greatest slope.
The point $C$ is $x$ metres from $A$, as shown in the diagram. The particle is released from rest at $C$ and moves up the plane.
\includegraphics{figure_8}
\begin{enumerate}[label=(\alph*)]
\item Show that, as the particle moves up the plane, the frictional force acting on the particle is 29.5 N, correct to three significant figures. [3 marks]
\item The particle comes to rest for an instant at $B$, which is 2 metres from $A$.
The particle then starts to move back towards $A$.
\begin{enumerate}[label=(\roman*)]
\item Find $x$. [8 marks]
\item Find the acceleration of the particle as it starts to move back towards $A$. [4 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA M2 2014 Q8 [15]}}