Question

CLASS ID#: (3) GIVEN (35 PTS): Two uniform disks A (m=10 kg) & B (my=6 kg) are fixed as one, pivoting about the frictionless joint at O. Two cylinders C (me=6 kg) and D (my=10 kg) are attached by ideal cords wound around the outer and inner disks respectively. The system of masses is released from rest and the cords do not slip. FIND: (A) (-10 Pts) The TIME REQUIRED (1) for the cylinder C to reach an upward speed of 0.5 m/s; (B) (-10 Pts) The corresponding ANGULAR VELOCITY (0) of the compound disk at time ir (C) (-15 Pts) The corresponding TENSIONS (T. & T.) in the cords connecting the two cylinders C & D. 10 m C 6 10 LED

Answer 1

о ne cords connecting the two cylinders C&D. ra 200 mm ITc (Т 1 Tc ПТЬ ac 17 г. Та, с | 10 р 6(38) N lo (9.91)

Total mass moment of inertia of uniform disk;

1o = marả + 5mbri

$\Rightarrow I_o = \frac{1}{2}(10)(0.200)^2 + \frac{1}{2}(6)(0.150)^2$

$\Rightarrow I_o =0.2675\;\;kg.m^2$

For cylinder C;

$\uparrow \sum F_y = m_ca_c$

$\Rightarrow T_c - 6(9.81)= m_c(\alpha \times r_A)$

$\Rightarrow T_c - 6(9.81)= 6(\alpha \times 0.200)$

$\Rightarrow T_c=58.86+ 1.2\alpha$

For cylinder D;

$\downarrow \sum F_y = m_Da_D$

=10(9.81) - Tp = трах в)

→ 10(9.81) - Td = 10(a x 0.150)

$\Rightarrow T_D = 98.1 - 1.5\alpha$

For pulley;

$(CW\;+)\;\;\;\;\sum M_o = I_o\alpha$

$\Rightarrow T_D r_B - T_C r_A = I_o\alpha$

$\Rightarrow ( 98.1 - 1.5\alpha)(0.150) - (58.86+ 1.2\alpha)(0.200)= 0.2675\alpha$

→ a= 4.01775 rad/s?

(a)

Acceleration of C is;

$a_c = \alpha \times r_A = 4.01775(0.20) = 0.80355\;\;m/s^2$

Time required will be;

$t = \frac{v-v_o}{a_C} = \frac{0.5-0}{0.80355}$

$\Rightarrow t =0.6222\;\;s$

...(Answer)

(b)

Angular velocity will be;

$\omega = \omega_o + \alpha t$

$\Rightarrow \omega = 0 + (4.01775) (0.6222)$

$\Rightarrow \omega =2.5\;\;rad/s$

...(Answer)

(c)

Tension in C;

$T_c=58.86+ 1.2\alpha$

$\Rightarrow T_c=58.86+ 1.2(4.01775)$

$\Rightarrow T_c=63.6813\;\;N$

...(Answer)

Tension in D;

$T_D = 98.1 - 1.5\alpha$

$\Rightarrow T_D = 98.1 - 1.5(4.01775)$

$\Rightarrow T_D = 92.0734\;\;N$

...(Answer)