Three masses m_1 = 1 kg, m_2 = 2 kg, and m_3 = 3 kg are situated on an inclined

Question

Three masses m_1 = 1 kg, m_2 = 2 kg, and m_3 = 3 kg are situated on an inclined plane (theta = 30 degree) and connected by two strings. If the system is stationary and the tension in the ropes are T_12 (rope connecting m_1 and m_2). and T_23 (rope connecting m_2 and m_3), find the static frictional f_1s, f_2s and f_3s on the respective masses, m_1, m_2 and m_3. If the system is moving down the inclined plane and the coefficients of kinetic friction are mu_k1 = 0.1, mu_k2 = 0.2, and mu_k3 = 0.3 for the respective masses, find the tensions T_12 and T_23 in the strings (treated as massless).

Explanation / Answer

a)    For mass m1

1 * 9.8 * sin30   - T12 - f1s   = 0

=>   f1s   =    4.9 - T12

         For mass m2

=>    2 * 9.8 * sin30   + T12 - f2s - T23   = 0

=>      f2s   =   9.8   + T12   - T23  

    For mass m3

=>    f3s   =   3 * 9.8 * sin30 + T23  

=>     f3s   =   14.7 + T23  

b)      For mass m1

=> a =   4.9 - T12 - 0.848

     For mass m2

=> a =     4.9 + 0.5*T12 - 1.697 - 0.5*T23

   For mass m3

=>   a   = 4.9 + 0.33*T23 - 2.546

=>    T12 =   1.1348 N

=>   T23 =    1.7065 N

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