A particle of mass 10 kg is connected to another particle of mass 6 kg by a taut light inelastic string which passes over a smooth light pulley at the edge of a rough horizontal table. The coefficient of friction, $\, \mu$, between the 10 kg mass and the table is $\, \frac{2}{5}$. The system is released from rest. (i) Show on separate diagrams the forces acting on each particle. (ii) Find the common acceleration of the particles. (iii) Find the tension in the string. (iv) Comment on the motion of the system if $\mu \geq \frac{2}{3}$. Masses of 5 kg and 3 kg are connected by a taut light inelastic string which passes over a light smooth pulley, as shown in the diagram. The 5 kg mass lies on a smooth plane inclined at $\, \theta \to \frac{4}{3}$ to the horizontal, where $\tan \alpha = \frac{3}{4}$. The 3 kg mass hangs vertically. The system is released from rest. Find (i) the common acceleration of the masses (ii) the tension in the string.

Leaving Cert Applied Maths Connected Particles: A particle of mass 10 kg is conn

A particle of mass 10 kg is connected to another particle of mass 6 kg by a taut light inelastic string which passes over a smooth light pulley at the edge of a rough horizontal table - Leaving Cert Applied Maths - Question 4 - 2017

Question 4

A particle of mass 10 kg is connected to another particle of mass 6 kg by a taut light inelastic string which passes over a smooth light pulley at the edge of a roug... show full transcript

Worked Solution & Example Answer:A particle of mass 10 kg is connected to another particle of mass 6 kg by a taut light inelastic string which passes over a smooth light pulley at the edge of a rough horizontal table - Leaving Cert Applied Maths - Question 4 - 2017

Step 1

Show on separate diagrams the forces acting on each particle.

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Answer

For the 6 kg mass, the forces are:

Tension, T, acting upwards.
Weight, W = 6g, acting downwards.
Frictional force, F_f = $\mu R_1$ , acting leftward, where $R_1$ is the normal force due to the table, which equals 6g.

For the 10 kg mass, the forces are:

Tension, T, acting upwards.
Weight, W = 10g, acting downwards.
Frictional force, F_f = $\mu R_2$ , acting leftward, where $R_2$ equals 10g.

Step 2

Find the common acceleration of the particles.

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Answer

Applying Newton’s second law to both masses:

For the 6 kg mass: $6g - T - \frac{2}{5}(6g) = 6a$ Rearranging gives: $T = 6g - \frac{2}{5}(6g) - 6a$

For the 10 kg mass: $T - 10g + \frac{2}{5}(10g) = 10a$ Substituting from the first equation, we can find: $T - 10g + 4g = 10a$ Solving these equations gives the common acceleration as: $a = \frac{5}{4} \, m/s^2$

Step 3

Find the tension in the string.

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Answer

Using the common acceleration: Substituting into one of the equations to find T: $T - 4g = 10a \implies T = 4g + 10a$ Plugging in values gives: $T = 40 + 12.5 = 52.5 \, N$

Step 4

Comment on the motion of the system if $\mu \geq \frac{2}{3}$.

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Answer

If $\mu \geq \frac{2}{3}$ , then the frictional force would be greater than the gravitational force acting on the 10 kg mass. This would result in no net motion in the system, hence the system would remain stationary.

Step 5

Find (i) the common acceleration of the masses.

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Answer

Applying Newton's second law to each mass:

For the 5 kg mass:

\sin(\alpha) = \frac{3}{5}.$$ This gives: $$3g - T = 5a\implies T = 3g - 5a.$$ For the 3 kg mass: $$5g - T = 3a$$ Thus, manipulating these equations leads to: The common acceleration calculated is: $$a = \frac{5}{4} \, m/s^2$$.

Step 6

Find (ii) the tension in the string.

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Answer

Using the values obtained: From previous equations: Substituting the common acceleration back into the equation for tension, we find: $T = 30 + 12.5 = 33.75 \, N.$ Therefore, the tension in the string is 33.75 N.

A particle of mass 10 kg is connected to another particle of mass 6 kg by a taut light inelastic string which passes over a smooth light pulley at the edge of a rough horizontal table - Leaving Cert Applied Maths - Question 4 - 2017

Worked Solution & Example Answer:A particle of mass 10 kg is connected to another particle of mass 6 kg by a taut light inelastic string which passes over a smooth light pulley at the edge of a rough horizontal table - Leaving Cert Applied Maths - Question 4 - 2017

Show on separate diagrams the forces acting on each particle.

Find the common acceleration of the particles.

Find the tension in the string.

Comment on the motion of the system if $\mu \geq \frac{2}{3}$.

Find (i) the common acceleration of the masses.

Find (ii) the tension in the string.

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