Question

An Atwood's machine consists of two masses, m1 and m2, connected by a string that passes over a pulley. If the pulley is a disk of radius R and mass M , find the acceleration of the masses. Express your answer in terms of the variables m1, m2, R, M, and appropriate constants.

Answer 1

Answer:

Explanation:

Suppose m₂ is greater than m₁ and it is going down . m₁ will be going up.

Let tension in string be T₁ and T₂  . Let common acceleration of system be a

For motion of m₁

T₁-m₁g = m₁a ----- (1)

For motion of m₂

m₂g- T₂ = m₂a ------- (2)

For motion of pulley

(T₂-T₁ )R represents net torque

(T₂-T₁ )R = I x α where I is moment of inertia of disc and α is angular acceleration of disc

(T₂-T₁ )R = 1/2 M R² x a / R

(T₂-T₁ ) = M a /2

Adding (1) and (2)

(m₂-m₁)g = (m₂+m₁)a + (T₂-T₁ )

(m₂-m₁)g = (m₂+m₁)a + Ma/2

(m₂-m₁)g = (m₂+m₁+ 0.5M)a

a = (m₂-m₁)g / ( (m₂+m₁+ 0.5M)

Answer 2

The acceleration of the masses is $\dfrac{g(m_2+m_1)}{(m_2-m_1-0.5M)}$.

Given to us

Masses = m₁, m₂

The radius of the pulley = R

Mass of the pulley = M

Assumption

• Let the mass m₁ > m₂. therefore, the mass m₁ is going down due to its weight while m₂ is going up.
• Assuming the tension in the string be T₁ and T₂, respectively.
• Also, the common acceleration in the system is a.

Tensions in strings

We know the acceleration due to gravity is denoted by g,

Tension in string 1, T₁

$T_1 = m_1(a+g)$......... equation 1

Tension in string 2, T₂

$T_2 = m_2(a-g)$......... equation 2

Inertia and acceleration of the pulley

The inertia of the pulley,  $I = \dfrac{1}{2}MR^2$

Acceleration of the pulley, $a = {\alpha }\times {R}$

Torque in the pulley

$(T_2-T_1)R = I \times \alpha$

Substitute the values we get,

$[m_2(a-g)-m_1(a+g)]R = \dfrac{1}{2} MR^2 \times \dfrac{a}{R}$

$[m_2a-m_2g-m_1a-m_1g]R= \dfrac{1}{2} MR \times a$

$m_2a-m_1a-m_2g-m_1g= \dfrac{1}{2} M \times a$

$a(m_2-m_1)-g(m_2+m_1)= 0.5M \times a$

$a(m_2-m_1)-0.5Ma=g(m_2+m_1)\\\\a(m_2-m_1-0.5M) = g(m_2+m_1)\\\\a = \dfrac{g(m_2+m_1)}{(m_2-m_1-0.5M)}$

Hence, the acceleration of the masses is $\dfrac{g(m_2+m_1)}{(m_2-m_1-0.5M)}$.

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