Two particles A and B, of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially B is held at rest on a rough fixed plane inclined at angle θ to the horizontal, where tan θ = $rac{5}{12}$. The part of the string from B to P is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, P, fixed at the top of the plane. The particle A hangs freely below P, as shown in Figure 4. The coefficient of friction between B and the plane is $rac{2}{3}$. The particles are released from rest with the string taut and B moves up the plane. (a) Find the magnitude of the acceleration of B immediately after release. (b) Find the speed of B when it has moved 1 m up the plane. (c) Find the time between the instances when the string breaks and when B comes to instantaneous rest.

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Two particles A and B, of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string - Edexcel - A-Level Maths Mechanics - Question 7 - 2011 - Paper 1

Question 7

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Two particles A and B, of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially B is held at rest on a rough fixed plan... show full transcript

Worked Solution & Example Answer:Two particles A and B, of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string - Edexcel - A-Level Maths Mechanics - Question 7 - 2011 - Paper 1

Step 1

Find the magnitude of the acceleration of B immediately after release.

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Answer

To find the acceleration, we analyze the forces acting on both particles A and B. For particle A:

The weight of A acts downwards (7g). Hence, the tension T in the string can be described by the equation:

$7g - T = 7a$
For particle B: The forces are the tension T acting upwards and the weight of B and the friction force acting down the plane. The equations are:

$T - F - 3g ext{sin}θ = 3a$
where F is the friction force and is given by:

$F = μR = rac{2}{3} R$
The normal force R acting on B is given by:

$R = 3g ext{cos}θ$
We can substitute R into the friction equation:

$F = rac{2}{3}(3g ext{cos}θ) = 2g ext{cos}θ$
Substituting this into the equation for B:

$T - 2g ext{cos}θ - 3g ext{sin}θ = 3a$
We can solve the equations together:

By eliminating T from the two equations:

$7g - (2g ext{cos}θ + 3g ext{sin}θ + 3a) = 7a$

Re-arranging gives:

$7g = 4a + 2g ext{cos}θ + 3g ext{sin}θ$
By substituting $an θ = rac{5}{12}$ , we find the values of sin and cos:

$ext{sin}θ = rac{5}{13}, ext{cos}θ = rac{12}{13}$
Finally, substituting into the equation gives:

$a = rac{2g}{5} ext{ or } 3.92 ext{ m/s}^2$ .

Step 2

Find the speed of B when it has moved 1 m up the plane.

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Answer

Using the equations of motion for B:

The initial velocity u = 0, the distance s = 1 m, and the acceleration a is already calculated as $rac{2g}{5}$ . The equation used here is:

$v^2 = u^2 + 2as$

Substituting our values:

$v^2 = 0 + 2 imes rac{2g}{5} imes 1$

Therefore:

ho{2g}{5} = 2.8 ext{ m/s}$$.

Step 3

Find the time between the instances when the string breaks and when B comes to instantaneous rest.

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Answer

When B has moved 1 m up the plane, the forces on B balance out:

The net force on B at this point is:

$-(F + 3g ext{sin}θ) = 3a$ Therefore:

$rac{2}{3} imes 3g ext{cos}θ + 3g ext{sin}θ = 3a$
We substitute the known values:

$rac{2}{3} imes 3g imes rac{12}{13} + 3g imes rac{5}{13} = 3(-a)$

Simplifying gives us:

$t = rac{2}{7} ext{ seconds} ext{ or } 0.286 ext{ seconds}$ .

Two particles A and B, of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string - Edexcel - A-Level Maths Mechanics - Question 7 - 2011 - Paper 1

Worked Solution & Example Answer:Two particles A and B, of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string - Edexcel - A-Level Maths Mechanics - Question 7 - 2011 - Paper 1

Find the magnitude of the acceleration of B immediately after release.

Find the speed of B when it has moved 1 m up the plane.

Find the time between the instances when the string breaks and when B comes to instantaneous rest.

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