From the particle under a net force model, apply Newton\'s second law to object

Question

From the particle under a net force model, apply Newton's second law to object 1: sigma F_y = T - m_1g = m_1a_y Apply Newton's second law to object 2: sigma F_y = m_2g - T = m_2a_y Add Equation (2) to Equation (1), noticing that T cancels: -m_1g + m_2g = m_1a_y + m_2a_y Solve for the acceleration (Use the following as necessary: m_, m_2, and g): a_y = (m_2 - m_1/m_1 + m_2)g Substitute Equation (3) into Equation (1) to find T (Use the following as necessary: m_1, m_2, and g): T = m_1(g + a_y) = (2m_2m_1/m_1+m_2)g Finalize The acceleration given by Equation (3) can be interpreted as the ratio of the magnitude of the unbalanced force on the system (m_2 - m_1)g to the total mass of the system (m_2 + m_1), as expected from Newton's second law. Notice that the sign of the acceleration depends on the relative masses of the two objects. Suppose that in the same Atwood setup another string is attached to the bottom of m_1 and a constant force f is applied, retarding the upward motion of m_1. If m_1 = 6.20 kg and m_2 = 11.40 kg, what value of f will reduce the acceleration of the system by 60%? ___________N

Explanation / Answer

acceleration from equation 3

a=[(11.4-6.2)/(11.4+6.2)]*9.8=2.9 m/s2

now acceleration is reduced by 60 % => a = 2.9(1-0.6)=1.158 m/s2
T = 6.2*(9.8+a) + f
=>T = 6.2*(9.8+1.158) + f=67.94+f
and

T + 11.4*a = 11.4*9.8

67.94+f=11.4(9.81-1.158)

f=30.6 N

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