Mathematics · Oct / Nov 2022 · Paper 51 · Question 7
YMA01
As printed in the paper
The question as text
7. Q (m) P (m) Figure 4 A particle P of mass m is attached to one end of a light inextensible string. Another particle Q, also of mass m, is attached to the other end of the 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 hangs vertically below the pulley with the string taut, as shown in Figure 4. The pulley, P and Q all lie in the same vertical plane. The coefficient of friction between Q and the table is μ, where μ < 1 Particle Q is released from rest. The tension in the string before Q hits the pulley is kmg, where k is a constant.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
7(a)[7]
F
mg
For P: mg
−
kmg
=
ma
Allow mg
−
T
=
ma
For Q: kmg
−
F
=
ma
Allow T
−
=
F
ma
2
Either of these may be replaced by :
mg
−
F
=
ma
(whole system)
Produce an equation in k and μonly using T = kmg
1 (1
)
k
=
+
μ
2
Attempt to find the acceleration.
1
7(b)[4]
[Note that some possible correct forms are:
a
=
g
(1
−
μ
)
or (1
g
−
k
)
2
or (
g k
−
μ
)
]
1
1
=
×
(1
−
μ
)
2
d
g
t
2
2
4
d
t
=
g
(1
−
μ
)
P or Q (or the system) would not move
7(c)[2]
Accept any of T = mg, T
>
,T
≥
, a = 0, a < 0,
0
mg
mg
a ≤
F = T, F > T, F ≥ T , F > mg. Allow F replaced by
μ
R
N.B. Forces referred to must be clearly defined so
e.g. use of vague terms like ‘forward force’ , ‘opposite force’, ‘force to
the left or right’ is B0.
Notes for question 7
B1 for F
=
μ
mg
seen e.g. on a diagram
7(a)
M1 Equation of motion for P with correct no. of terms, condone sign
errors
A1 Correct equation (allow a)
M1 Equation of motion for Q with correct no. of terms, condone sign
errors
A1 Correct equation (allow a)
N.B. (a) must be used in both equations
M1 for producing an equation in k and μonly
A1 oe Must appear in (a)
M1 Attempt to find the acceleration in terms of g and μ or g and k or g,
7(b)
k and μ
M1 Complete method to find an equation in d, g, t and μonly, condone a
sign error.
A1 Correct equation in d, g, t and μonly
A1 Any equivalent form
B1 Correct statement. B0 if incorrect extras.
7(c)
DB1 Correct reason
Allow column vectors throughout
2
2
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