Question

A light, inextensible cord passes over a light, frictionless pulley with a radius of 8.9 cm. It has a 19 kg mass on the left and a 2.6 kg mass on the right, both hanging freely. Initially their center of masses are a vertical distance 3.8 m apart. The acceleration due to gravity is 9.8 m/s^2. At what rate are the two masses accelerating when they pass each other? Answer in units of m/s^2. What is the tension in the cord when they pass each other? Answer in units of N.

Answer 1

Answer

given,

m₁ = 19 kg

m₂ = 2.6 kg

radius = 8.9 cm

vertical distance = 3.8 m

acceleration due to gravity = 9.8 m/s²

a) $a = \dfrac{F}{m}$

$a = \dfrac{g\times (m_1-m_2)}{m_1+m_2}$

$a = \dfrac{9.8\times (19-2.6)}{19.6+2.6}$

a = 7.24 m/s²

b) T = 2.6 × (9.8 + 7.24)

T = 44.30 N

Answer 2

A light, inextensible cord passes over a light, frictionless pulley with a radius of 8.9 cm. It has a 19 kg mass on the left and a 2.6 kg mass on the right, both hanging freely. The rate at which the two masses are accelerating when they pass each other is 7.44 m/s². The tension in the cord is 44.84 N.

In a light inextensible cord;

• The mass on the right side m₁ = 19 kg
• The mass on the left side m₂ = 2.6 kg
• The radius of the pulley r = 0.089 m

Using Newton's Law to the mass m₁

m₁ g - T = m₁ a

where;

• the tension T = m₁ (g -a) ----- Let this be equation (1)

For mass m₂,  

• T - m₂ g = m₂a   ------- Let this be equation (2)

equation both equation (1) and (2) together;

$\mathbf{m_1(g-a) -m_2g = m_2a}$

$\mathbf{m_1g-m_1a -m_2g = m_2a}$

By rearrangement and collecting like terms;

a(m₂ + m₁) = g(m₁ - m₂)

Making acceleration (a) the subject of the formula:

$\mathbf{a = \dfrac{g(m_1-m_2)}{(m_2+m_1)}}$

$\mathbf{a = \dfrac{9.8(19-2.6)}{(2.6+19)}}$

$\mathbf{a = \dfrac{9.8(16.4)}{(21.6)}}$

a = 7.44 m/s²

From equation (1), the tension in the cord is:

Tension (T) = m₁ (g -a)

T = 19(9.8 - 7.44)

T = 19(2.36)

T = 44.84 N

Therefore, we can conclude that the rate at which the two masses accelerate when they pass each other is 7.44 m/s², and tension in the cord when they pass each other is 44.84 N

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