Question

Two masses —m1 and m2— are connected by light cables to the perimeters of two cylinders of radii r1 and r2 respectively, as shown in the diagram. The cylinders are rigidly connected to each other on a common axle. The moment of inertia of the pair of cylinders is =45 kgm^2. Also, r1= 0.5 m, r2 = 1.5 m, and m1 = 20 kg. Note: m1 is hanging from the small cylinder, and m2 is hanging from the large cylinder. Determine m2 such that the system will remain in equilibrium. The mass m2 is removed and the system is released from. Determine the angular acceleration of the cylinders. Determine the tension in the cable supporting m1. Determine the linear speed of m1 at the time it has descended 1 m.

Answer 1

Answer:

Part a)

Mass of m2 is given as

$m_2 = \frac{20}{3} kg$

Part b)

Angular acceleration is given as

$\alpha = 1.96 rad/s^2$

Part c)

Tension in the rope is given as

$T = 176.6 N$

Explanation:

Part a)

When m1 and m2 both connected to the cylinder then the system is at rest

so we can use torque balance here

$m_1g r_1 = m_2 g r_2$

$20 g(0.5) = m_2 g(1.5)$

$10 = 1.5 m_2$

$m_2 = \frac{20}{3} kg$

Part b)

When block m_2 is removed then system becomes unstable

so force equation of mass m1

$m_1g - T = m_1 a$

also we have

$T r_1 = I\alpha$

now we have

$m_1g = \frac{I a}{r_1^2} + m_1 a$

$a = \frac{m_1g}{\frac{I}{r_1^2} + m_1}$

$a = \frac{20 (9.81)}{\frac{45}{0.5^2} + 20}$

$a = 0.981 m/s^2$

so angular acceleration is given as

$\alpha = \frac{a}{r_1}$

$\alpha = \frac{0.981}{0.5}$

$\alpha = 1.96 rad/s^2$

Part c)

Tension in the rope is given as

$T = \frac{I\alpha}{r_1}$

$T = \frac{45 (1.96)}{0.5}$

$T = 176.6 N$

Answer 2

a). The m₂ to keep in equilibrium is 20/3 Kg

b). The angular acceleration = 1.96 rad/s²

c). Tension = 176.6 N

d). t is not given to find t²

Find the angular speed

What information do we have:

Masses m₁ and m₂

Moment of inertia = 45 kgm²

r¹ = 0.5 m

r² = 1.5 m

m¹ = 20 kg

a). To find,

Mass of m² = ?

we know,

m₁ gr¹ = m₂gr₂

⇒ 20(0.5) = m₂g(1.5)

∵ m₂ = 20/3 Kg

b). Angular acceleration can be denoted by:

m₁ g - T = m₁ a

using this formula, we can find:

m₁ g = Ia/$r^{2}_{1}$ + m₁ a

⇒ a = 20.981/(45/0.5² + 20)

∵ a $= 0.981 m/s$²

Now,

Angular acceleration = a/r¹

= 0.981/0.5

= 1.96 rad/s²

c). Tension in the cable:

T = Ia/r¹

= 45(1.96)/0.5

= 176.6 N

d). Linear speed can't be determined as t is not provided.

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