| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2014 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Small oscillations with elastic strings/springs |
| Difficulty | Challenging +1.2 This is a standard M4 energy methods question requiring systematic application of elastic PE and gravitational PE formulas, followed by routine equilibrium analysis via differentiation. The geometry (finding extension using cosine rule) and stability determination are textbook techniques, though the algebra requires care across multiple steps. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.04e Rigid body equilibrium: coplanar forces |
Total
TOTAL FOR PAPER: 75 MARKS
Question 7:
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1. A particle A has constant velocity (3i+j)ms–1 and a particle B has constant
velocity (i–k)ms–1. At time t = 0 seconds, the position vectors of the particles A and B
with respect to a fixed origin O are (–6i+4j–3k)m and (–2i+2j+3k)m respectively.
(a) Show that, in the subsequent motion, the minimum distance between A and B
is 4(cid:165)2m.
(6)
(b) Find the position vector of A at the instant when the distance between A and B is a
minimum.
(2)
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2. A car of mass 1000kg is moving along a straight horizontal road. The engine of the car
is working at a constant rate of 25kW. When the speed of the car is vms–1, the resistance
to motion has magnitude 10v newtons.
(a) Show that, at the instant when v = 20, the acceleration of the car is 1.05ms–2.
(3)
(b) Find the distance travelled by the car as it accelerates from a speed of 10ms–1 to a
speed of 20ms–1.
(8)
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3. A small ball is moving on a smooth horizontal plane when it collides obliquely with
a smooth plane vertical wall. The coefficient of restitution between the ball and the
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wall is . The speed of the ball immediately after the collision is half the speed
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of the ball immediately before the collision.
Find the angle through which the path of the ball is deflected by the collision.
(8)
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4. At noon two ships A and B are 20 km apart with A on a bearing of 230(cid:113) from B. Ship B is
moving at 6 km h–1 on a bearing of 015(cid:113). The maximum speed of A is 12 km h–1. Ship A
sets a course to intercept B as soon as possible.
(a) Find the course set by A, giving your answer as a bearing to the nearest degree.
(4)
(b) Find the time at which A intercepts B.
(4)
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Question 4 continued
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5.
u
A(m)
(cid:533)
(cid:302)
B(3m)
3u
Figure 1
Two smooth uniform spheres A and B have equal radii. The mass of A is m and the mass
of B is 3m. The spheres are moving on a smooth horizontal plane when they collide
obliquely. Immediately before the collision, A is moving with speed 3u at angle (cid:302) to the
line of centres and B is moving with speed u at angle (cid:533) to the line of centres, as shown in
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Figure 1. The coefficient of restitution between the two spheres is . It is given that
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cos(cid:302)= and cos(cid:533)= and that (cid:302) and (cid:533)(cid:3)are both acute angles.
3 3
(a) Find the magnitude of the impulse on A due to the collision in terms of m and u.
(8)
(b) Express the kinetic energy lost by A in the collision as a fraction of its initial kinetic
energy.
(4)
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Question 5 continued
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6. A particle of mass mkg is attached to one end of a light elastic string of natural length
ametres and modulus of elasticity 5ma newtons. The other end of the string is attached
to a fixed point O on a smooth horizontal plane. The particle is held at rest on the plane
with the string stretched to a length 2a metres and then released at time t=0. During
the subsequent motion, when the particle is moving with speed vms–1, the particle
experiences a resistance of magnitude 4mv newtons. At time t seconds after the particle
is released, the length of the string is (a+x) metres, where 0(cid:45)x(cid:45)a.
(a) Show that, from t = 0 until the string becomes slack,
d2x dx
+ 4 + 5x = 0
dt2 dt
(3)
(b) Hence express x in terms of a and t.
(6)
(c) Find the speed of the particle at the instant when the string first becomes slack, giving
your answer in the form ka, where k is a constant to be found correct to 2 significant
figures.
(4)
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7.
A
(cid:537)
r
(cid:85)(cid:3)+ x
O
B
mg
Figure 2
A bead B of mass m is threaded on a smooth circular wire of radius r, which is fixed in
a vertical plane. The centre of the circle is O, and the highest point of the circle is A.
A light elastic string of natural length r and modulus of elasticity kmg has one end
attached to the bead and the other end attached to A. The angle between the string and the
downward vertical is (cid:537), and the extension in the string is x, as shown in Figure 2.
Given that the string is taut,
(a) show that the potential energy of the system is
2mgr{(k – 1)cos2(cid:537)(cid:3)(cid:177)(cid:3)(cid:78)cos(cid:537)} + constant
(6)
Given also that k=3,
(b) find the positions of equilibrium and determine their stability.
(9)
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Question 7 continued
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(Total 15 marks)
TOTAL FOR PAPER: 75 MARKS
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*P43159A02424*\includegraphics{figure_2}
A bead $B$ of mass $m$ is threaded on a smooth circular wire of radius $r$, which is fixed in a vertical plane. The centre of the circle is $O$, and the highest point of the circle is $A$. A light elastic string of natural length $r$ and modulus of elasticity $kmg$ has one end attached to the bead and the other end attached to $A$. The angle between the string and the downward vertical is $\theta$, and the extension in the string is $x$, as shown in Figure 2.
Given that the string is taut,
\begin{enumerate}[label=(\alph*)]
\item show that the potential energy of the system is
$$2mgr[(k-1)\cos^2 \theta - k\cos \theta] + \text{constant}$$ [6]
\end{enumerate}
Given also that $k = 3$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the positions of equilibrium and determine their stability. [9]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2014 Q7 [15]}}