Mathematics · Oct / Nov 2023 · Paper 51 · Question 7
YMA01
As printed in the paper
The question as text
7. A (m) B (km) α Figure 4 Figure 4 shows a block A of mass m held at rest on a rough plane. The plane is inclined at an angle α to the horizontal and the coefficient of friction between the block and the plane is μ. One end of a light inextensible string is now attached to A. The string passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a block B of mass km. Block B hangs vertically below the pulley, with the string taut. The string from A to the pulley lies along a line of greatest slope of the plane. Both A and B are modelled as particles. When the system is released from rest, A moves up the plane and the tension in the mg string is 4 3 7 Given that μ = 1 3 and tan α = 24
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
7(a)[9]
For A: 4
mg
sin
−
mg
α
−
F
=
ma
3
cos
R
=
mg
α
1
Use of
=
in an equation.
F
R
3
11
g
a =
or 0.73g or better
15
(ii)
4
mg
For B:
−
=
kmg
kma
3
k =
5
N.B. Either equation of motion could be replaced by a whole
system equation:
−
sin
α
−
=
(
+
1)
kmg
mg
F
k
ma
Complete method to find resultant force
90
°−
α
2 cos
T
2
Substitute T = 4
mg and trig
3
32
mg or 2.1mg or better.
15
ALT 1
Use of cosine rule:
2
+
2
−
2( )( )cos(90
+
α
)
T
T
T T
ALT 2
Use of vert and horiz components to find the resultant:
( cos
α
)
2
+
(
+
sin
α
)
2
T
T
T
Notes for question 7
For A use F=ma parallel to the plane. Must be dimensionally correct and
(a) M1
have correct no of terms. Condone sin/cos confusion.
N.B. If they use T in this equation and never replace it, allow M1.
A1
Correct unsimplified equation.
N.B. a could be replaced by a
Resolve perpendicular to the plane Must be dimensionally correct and
M1
have correct no of terms. Condone sin/cos confusion.
Correct equation
A1
1
M1
Use of
F
=
R
3
A1
For B use F=ma vertically. Must be dimensionally correct and have
M1
correct no of terms. Condone sin/cos confusion.
N.B. Must have km on both sides for this mark.
N.B. If they use T in this equation and never replace it, allow M1.
Correct unsimplified equation
A1
N.B. a could be replaced by a, but must be consistent with the equation
for A.
A1
confusion
Correct expression
A1
Substitute T = 4
mg and trig, dependent on previous M mark
dM1
3
Correct answer.
A1
(b) ALT1
Complete method – must involve αor its numerical value
M1
A1
Correct expression
Substitute T = 4
mg and trig, dependent on previous M mark
dM1
3
1024
2
2
m g
A1
Correct answer. Allow
or similar.
225
(b) ALT2
M1
Complete method, allow sin/cos confusion
Correct expression
A1
Substitute T = 4
mg and trig, dependent on previous M mark
dM1
3
1024
m g
2
2
A1
Correct answer. Allow
or similar.
225
Pearson Education Limited. Registered company number 872828
with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
7(b)[4]
Complete method to find resultant force
90
°−
α
2 cos
T
2
Substitute T = 4
mg and trig
3
32
mg or 2.1mg or better.
15
ALT 1
Use of cosine rule:
2
+
2
−
2( )( )cos(90
+
α
)
T
T
T T
ALT 2
Use of vert and horiz components to find the resultant:
( cos
α
)
2
+
(
+
sin
α
)
2
T
T
T
Notes for question 7
For A use F=ma parallel to the plane. Must be dimensionally correct and
(a) M1
have correct no of terms. Condone sin/cos confusion.
N.B. If they use T in this equation and never replace it, allow M1.
A1
Correct unsimplified equation.
N.B. a could be replaced by a
Resolve perpendicular to the plane Must be dimensionally correct and
M1
have correct no of terms. Condone sin/cos confusion.
Correct equation
A1
1
M1
Use of
F
=
R
3
A1
For B use F=ma vertically. Must be dimensionally correct and have
M1
correct no of terms. Condone sin/cos confusion.
N.B. Must have km on both sides for this mark.
N.B. If they use T in this equation and never replace it, allow M1.
Correct unsimplified equation
A1
N.B. a could be replaced by a, but must be consistent with the equation
for A.
A1
confusion
Correct expression
A1
Substitute T = 4
mg and trig, dependent on previous M mark
dM1
3
Correct answer.
A1
(b) ALT1
Complete method – must involve αor its numerical value
M1
A1
Correct expression
Substitute T = 4
mg and trig, dependent on previous M mark
dM1
3
1024
2
2
m g
A1
Correct answer. Allow
or similar.
225
(b) ALT2
M1
Complete method, allow sin/cos confusion
Correct expression
A1
Substitute T = 4
mg and trig, dependent on previous M mark
dM1
3
1024
m g
2
2
A1
Correct answer. Allow
or similar.
225
Pearson Education Limited. Registered company number 872828
with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
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