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Suppose the Rod in the Previous Problem Has a Mass of 1 Kg Distributed Uniformly Over Its Length. (A) Find the Initial Angular Acceleration of the Rod. - Physics

Sum

Suppose the rod in the previous problem has a mass of 1 kg distributed uniformly over its length.

(a) Find the initial angular acceleration of the rod.

(b) Find the tension in the supports to the blocks of mass 2 kg and 5 kg.

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Solution

Total moment of inertia of the system about the axis of rotation,

\[I_{net}  = \left( m_1 r_1^2 + m_2 r_2^2 + \frac{m l^2}{12} \right)\]

m and l are the mass and length of the rod, respectively.

\[ \tau_{net}  =  F_1  r_1  -  F_2  r_2 \]

\[\text{Also, }\tau_{net}  =  I_{net}  \times \alpha\]

On equating the value of \[\tau_{net}\] and putting the value of `l_("net"),` we get

\[F_1  r_1  -  F_2  r_2  = \left( m_1 r_1^2 + m_2 r_2^2 + \frac{m l^2}{12} \right) \times \alpha\]

\[\left( - 2 \times 10 \times 0 . 5 \right) + \left( 5 \times 10 \times 0 . 5 \right) = \left[ 5 \left( \frac{1}{2} \right)^2 + 2 \left( \frac{1}{2} \right)^2 + \frac{\left( 1 \right)^2}{12} \right]  \alpha\]

\[\Rightarrow   15 = \left( 1 . 75 + 0 . 084 \right)  \alpha\]

\[ \Rightarrow \alpha = \frac{1500}{\left( 175 + 8 . 4 \right)} = \frac{1500}{183 . 4}\]

\[= 8 . 1\text{ rad/s}^2   ........\left( g = 10 \right)\]

\[= 8 . 01\text{ rad/s}^2   ......\left(\text{if }g = 9 . 8 \right)\]

(b) From the free body diagram of the block of mass 2 kg,

\[T_1  -  m_1 g =  m_1 a\] 

\[ \Rightarrow  T_1  = 2  \left( a + g \right)\] 

\[ = 2\left( \alpha r + g \right)........ \left(\text{using, }a = \alpha r \right)\] 

\[ = 2\left( 8 \times 0 . 5 + 9 . 8 \right)\] 

\[ \Rightarrow  T_1  = 27 . 6  N\]

From the free body diagram of the block of mass 5 kg,

\[m_2 g -  T_2  =  m_2 a\]

\[ \Rightarrow  T_2  =  m_2 \left( g - a \right)\]

\[ = 5  \left( g - a \right) = 5  \left( 9 . 8 - 8 \times 0 . 5 \right)............\left( a = \alpha r \right)\]

\[= 5 \times 5 . 8 = 29  N\]

Concept: Values of Moments of Inertia for Simple Geometrical Objects (No Derivation)
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APPEARS IN

HC Verma Class 11, 12 Concepts of Physics 1
Chapter 10 Rotational Mechanics
Q 32 | Page 197
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