One end of a light inextensible string is attached to a block P of mass 5 kg. The block P is held at rest on a smooth fixed plane which is inclined to the horizontal at an angle α, where sin α = rac{3}{5}. The string lies along a line of greatest slope of the plane and passes over a smooth light pulley which is fixed at the top of the plane. The other end of the string is attached to a light scale pan which carries two blocks Q and R, with block Q on top of block R, as shown in Figure 3. The mass of block Q is 15 kg and the mass of block R is 10 kg. The scale pan hangs at rest and the system is released from rest. By modelling the blocks as particles, ignoring air resistance and assuming the motion is uninterrupted, find (a) (i) the acceleration of the scale pan, (ii) the tension in the string, (8) (b) the magnitude of the force exerted on block Q by block R, (3) (c) the magnitude of the force exerted on the pulley by the string.

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One end of a light inextensible string is attached to a block P of mass 5 kg - Edexcel - A-Level Maths Mechanics - Question 7 - 2009 - Paper 1

Question 7

One end of a light inextensible string is attached to a block P of mass 5 kg. The block P is held at rest on a smooth fixed plane which is inclined to the horizontal... show full transcript

Worked Solution & Example Answer:One end of a light inextensible string is attached to a block P of mass 5 kg - Edexcel - A-Level Maths Mechanics - Question 7 - 2009 - Paper 1

Step 1

(a) (i) the acceleration of the scale pan

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Answer

To find the acceleration of the scale pan, we apply Newton's second law. Here, the tension T in the string acts on block P while the gravitational force pulls it down the incline.

Using the equation:
$T - 5g imes ext{sin}( rac{3}{5}) = 5a$
we know that the weight of the combined blocks Q and R gives us another equation:
$15g - T = 15a$
Substituting T from the first equation into the second, we can solve for a.
Following through:
$15g - (5a + 5g imes ext{sin}( rac{3}{5})) = 15a$
This gives us the final formula to calculate acceleration. After rearranging and substituting known values, we find:

$a = 0.6g$
Therefore, the acceleration of the scale pan is 0.6g.

Step 2

(a) (ii) the tension in the string

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Answer

With the acceleration known, we substitute back to find the tension T. From the first derived equation:

$T = 5g imes ext{sin}( rac{3}{5}) + 5a$
We can now use the value of a found previously:

$T = 15a$ which leads to:
$T = 6g.$
Consequently, the tension in the string is 6g.

Step 3

(b) the magnitude of the force exerted on block Q by block R

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Answer

To find the force exerted on block Q by block R, we apply Newton's second law again. The force acting downwards on Q is its weight, given by:

$F_Q = m_Q imes g = 15g.$
The normal force acting on block Q, due to block R, is calculated with:

$N_Q = 5a, ext{ where } a ext{ is the acceleration calculated in part (a).}$
Thus, substituting the values:

$N_Q = 5a = 5(0.6g) = 3g.$
Therefore, the magnitude of the force exerted on block Q by block R is 3g.

Step 4

(c) the magnitude of the force exerted on the pulley by the string

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Answer

For this part, we can analyze the forces acting on the pulley. The forces exerted by the string are reflected by T. This leads us to use the equation:

$F = 2T imes ext{cos}(90° - α)$
Substituting known values, where α has been determined and the necessary calculations yield:

$F = 2(6g) imes ext{cos}(26.56°)$
This resolves to:

$F = 12g imes ext{cos}(26.56°) = 105N.$
Thus, the magnitude of the force exerted on the pulley by the string is 105 N.

One end of a light inextensible string is attached to a block P of mass 5 kg - Edexcel - A-Level Maths Mechanics - Question 7 - 2009 - Paper 1

Worked Solution & Example Answer:One end of a light inextensible string is attached to a block P of mass 5 kg - Edexcel - A-Level Maths Mechanics - Question 7 - 2009 - Paper 1

(a) (i) the acceleration of the scale pan

(a) (ii) the tension in the string

(b) the magnitude of the force exerted on block Q by block R

(c) the magnitude of the force exerted on the pulley by the string

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