A crane is lifting a 180 kg crate upward with an acceleration of 1.2 m/s^2. The

Question

A crane is lifting a 180 kg crate upward with an acceleration of 1.2 m/s^2. The cable passes over a solid cylindrical pulley at the top of the boom. The pulley has a mass of 130 kg. The cable is then wound around a hollow cylindrical drum mounted on the deck of the crane. The mass of the drum is 150 kg and its radius is 0.76 m. The engine applies a counterclockwise torque to the drum in order to wind up the cable. What is the magnitude of the torque? (Ignore the mass of the cable) Hint: Use Newton's 2nd law in both it's linear form and sums of torque forms as after all you do have acceleration.

Explanation / Answer

Mass of the crate is m = 180 kg Mass of the pulley is M_p   = 130kg Mass of the drum is M_d = 150 kg Acceleration of the crate is a = 1.2 m/s^2 radius r = 0.76m Let T_1 and T_2 be the tension of the cable and crate Applying Newtons law for the pulley motion is T_1 r_2 - T_2 r_1 = I_2 _2^2 ........(1) Applying Newtons second law for drums rotation I_d _d = _d -T_2 R .........(2) 1/2 m_p r^2* a / r = (T_2 - T_1 ) r _d = T_2 R + I_2 _2^2         = m ( a + g ) R + 1/2 (m_p + m_d ) a R         = 180 kg ( 1.2 + 9.8 m/s^2 ) * 0.76m + 1/2 ( 150 kg + 130 kg ) (1.2m/s^2 ) ( 0.76 m )         = 1.632*10^3 N m                               

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