Two particles of masses 0-4 kg and 0-3 kg are attached to the ends of a light inextensible string which passes over a light smooth fixed pulley. They are held at the same level, as shown in the diagram. The system is released from rest. Find (i) the tension in the string (ii) the speed of the 0.4 kg mass when it has descended 0.7 m. A smooth wedge of mass 3m rests on a smooth horizontal surface. One face of the wedge makes an angle of 45° with the horizontal. A particle of mass m is placed on the smooth inclined face of the wedge. The system is released from rest. (i) Show, on separate diagrams, the forces acting on the wedge and on the particle. The particle moves with acceleration p relative to the wedge and the wedge moves with acceleration q. (ii) Find the value of p and the value of q.

Leaving Cert Applied Maths Connected Particles: Two particles of masses 0-4 kg a

Two particles of masses 0-4 kg and 0-3 kg are attached to the ends of a light inextensible string which passes over a light smooth fixed pulley - Leaving Cert Applied Maths - Question 4 - 2019

Question 4

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Two particles of masses 0-4 kg and 0-3 kg are attached to the ends of a light inextensible string which passes over a light smooth fixed pulley. They are held at the... show full transcript

Worked Solution & Example Answer:Two particles of masses 0-4 kg and 0-3 kg are attached to the ends of a light inextensible string which passes over a light smooth fixed pulley - Leaving Cert Applied Maths - Question 4 - 2019

Step 1

Find (i) the tension in the string

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Answer

To find the tension in the string, we start by analyzing the forces acting on the two masses. For the 0.4 kg mass, the equation of motion can be expressed as:

$T - 0.4g = 0.4a$

For the 0.3 kg mass:

$0.3g - T = 0.3a$

Adding both equations together:

$0.3g - 0.4g = 0.4a + 0.3a$

This simplifies to:

$-0.1g = 0.7a$

Thus, we can solve for acceleration a:

$a = -\frac{0.1g}{0.7} = 1.4 \, m/s^2$

Now substituting a back into the equation for tension:

T = 0.3(9.8) - 0.42\ T = 3.36 \, N$$

Step 2

Find (ii) the speed of the 0.4 kg mass when it has descended 0.7 m

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Answer

To find the speed, we can use the kinematic equation:

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

where:

$u = 0$ (initial speed)
$a = 1.4 \, m/s^2$ (acceleration)
$s = 0.7 \, m$ (distance)

Substituting the values:

v^2 = 1.96\ v = \sqrt{1.96} = 1.4 \, m/s$$

Step 3

Find (i) Show, on separate diagrams, the forces acting on the wedge and on the particle

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Answer

For the wedge:

Normal force R acts perpendicular to the surface.
Weight of the wedge: $3mg$ acts vertically downwards.

For the particle:

R acts along the inclined face of the wedge.
Weight $mg$ acts vertically downwards.
The acceleration of the wedge is q, and for the particle, it is p.

Step 4

Find (ii) the value of p and the value of q

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Answer

From the forces acting on the particle, we have:

$R \cos 45° = 3mq$ Substituting for R we find that: $R = 3\sqrt{2}mq$

From the wedge:

\Rightarrow mg \frac{1}{\sqrt{2}} = mq \frac{1}{\sqrt{2}}\ &mg - 6mq = mq\n\Rightarrow 6m = \frac{q}{1.4}$$ Thus, combining these expressions, we get: $$p = \frac{q}{\sqrt{2}}$$ Substituting the numeric values, we find: $$q = 7.92$$

Two particles of masses 0-4 kg and 0-3 kg are attached to the ends of a light inextensible string which passes over a light smooth fixed pulley - Leaving Cert Applied Maths - Question 4 - 2019

Worked Solution & Example Answer:Two particles of masses 0-4 kg and 0-3 kg are attached to the ends of a light inextensible string which passes over a light smooth fixed pulley - Leaving Cert Applied Maths - Question 4 - 2019

Find (i) the tension in the string

Find (ii) the speed of the 0.4 kg mass when it has descended 0.7 m

Find (i) Show, on separate diagrams, the forces acting on the wedge and on the particle

Find (ii) the value of p and the value of q

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