Mathematics · May / June 2024 · Paper 51 · Question 8
YMA01
As printed in the paper
The question as text
8. P B (3m) A (2m) h α Figure 4 One end of a light inextensible string is attached to a particle A of mass 2m. The other end of the string is attached to a particle B of mass 3m. Particle A is held at rest on a rough plane which is inclined to horizontal ground at an angle α, where tan 5 12 The string passes over a small smooth pulley P which is fixed at the top of the plane. Particle B hangs vertically below P with the string taut, at a height h above the ground, as shown in Figure 4. The part of the string between A and P lies along a line of greatest slope of the plane. The two particles, the string and the pulley all lie in the same vertical plane. The coefficient of friction between A and the plane is 11 36 The particle A is released from rest and begins to move up the plane.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
8(a)[3]
R
mg
11
12
22
mg
=
×
2
×
=
*
F
mg
36
13
39
8(b)[2]
3
−
=
3
mg
T
ma
8(c)[4]
22
mg
4
mg
−
−
2
sin
α
=
2
(
−
=
2
)
T
mg
ma
T
ma
39
3
22
mg
OR:
3
−
−
2
sin
α
=
5
mg
mg
ma
39
Solve for a in terms of g
N.B. Must reach a = kg from their equations
a = 1
3 g *
2
gh
8(d)[6]
v =
2
3
22
mg
2
sin
2
OR PE Gain = 2
sin
α
−
−
mg
α
= ±
ma
mgd
39
2
= 10
g
mgd
=
±
a
3
13
2
gh
2
g
0
2
=
−
×
×
d
3
3
22
1
2
10
mgd
gh
mgd
=
×
2
m
×
−
39
2
3
13
1
1
=
=
d
h
d
h
2
2
Total distance = 1
3
+
=
h
h
h
2
2
Notes
N.B. For (a) and (c), if fractions for the trig ratios are not seen or
implied, only penalise once.
M1: Resolve perp to the plane, correct terms, condone cos/sin confusion
8(a)
and sign errors. Allow if they use
cos(
12
) or similar.
13
A1: Correct equation
A1*: Correct given answer correctly obtained, must see 12
13
8(b)
M1:Equation of motion for B, correct terms, condone sign errors
A1: Correct equation
8(c)
M1:Equation of motion for A OR whole system, correct terms but
allow F, condone sign errors and sin/cos confusion.
Allow if they
use
sin(
13)
5
or similar.
A1: Correct equation
M1: Solve for a in terms of g, need to see trig substituted and must be
solving 2 equations in T and a OR using a whole system equation with
correct terms.
A1*: Correct given answer correctly obtained, must see 5
13
N.B. Allow a full verification, using the equations of motion for A and
B OR the whole system equation.
8(d)
B1: seen or implied
M1:Equation of motion for A, correct terms, condone sign errors and
sin/cos confusion
A1: Correct acceleration or deceleration of A
M1: Complete method to find an equation in d, g and h only, using a
new calculated a
A1: cao
A1ft: Their d (which must be a multiple of h) + h.
N.B. This mark is only dependent on the previous M.
OR: Using Workenergy
B1: cao
M1: PE gain of A, condone sign errors and sin/cos confusion
A1: Correct PE gain
M1: Use of workenergy principle to obtain an equation in m, d, g and h
only, using their PE expression
A1: cao
A1ft: Their d (which must be a multiple of h) + h, with the final answer
of the form kh.
Pearson Education Limited. Registered company number 872828
with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
Make a free account to save your answers and see your weak topics and ask the AI why a wrong answer is wrong.