Chapter 8, Problem 31P

(a)

To determine

To Find: The moment of inertia based on vertical axis.

Answer to Problem 31P

  $I=6.125\text{kg}\cdot {\text{m}}^{\text{2}}$

Explanation of Solution

Given information:

  $\begin{array}{l}m=1.8\text{kg}\\ \text{M}=\text{3}\text{.1}\text{}\text{kg}\end{array}$

The objects are wired together by very light, rigid pieces of wire.

Formula used:

  $I=m{r}^2+M{R}^2+M{r}^2+m{R}^2$

Here,

  $I$ is the moment of inertia,

  $m,M$ are the mass of the objects,

  $r,R$ is the distance of objects from axis.

Calculation:

In the figure $m=1.8\text{kg}\text{and }M=3.1\text{kg}$

And as the objects are wired together by very light rigid pieces of wire, so their mass can be neglected.

When the axis of rotation is vertical

Moment of inertial about Y axis will be

  $\begin{array}{l}I=m{r}^2+M{R}^2+M{r}^2+m{R}^2\\ =m\left(r^2+R^2\right)+M\left(r^2+R^2\right)\\ =1.8\left({0.5}^2+1^2\right)+3.1\left({0.5}^2+1^2\right)\\ =\left(2.25+3.875\right)\text{kg}\cdot {\text{m}}^{\text{2}}\\ I=6.125\text{kg}\cdot {\text{m}}^{\text{2}}\end{array}$

Conclusion:

  $I=6.125\text{kg}\cdot {\text{m}}^{\text{2}}$

(b)

To determine

To find: The moment of inertia based on horizontal axis.

Answer to Problem 31P

The value is $I=0.61\text{kg}\cdot {\text{m}}^{\text{2}}$

Explanation of Solution

Given information:

  $\begin{array}{l}m=1.8\text{kg}\\ \text{M}=\text{3}\text{.1}\text{}\text{kg}\end{array}$

The objects are wired together by very light, rigid pieces of wire.

Formula used:

  $I={\displaystyle \sum M_1{R}_1{}_y^2}$

Here,

  $I$ is the moment of inertia,

  $M$ is the mass of the object,

  $R$ is the distance of objects from axis.

Calculation:

The moment of inertia about the X axis

  $\begin{array}{l}I={\displaystyle \sum M_1{R}_1{}_y^2}\\ =\left[\left(2m+2M\right){\left(0.25\right)}^2\right]\text{kg}\cdot {\text{m}}^{\text{2}}\\ =\left[\left(2\times 1.8+2\times 3.1\right){\left(0.25\right)}^2\right]\text{kg}\cdot {\text{m}}^{\text{2}}\\ I=0.61\text{kg}\cdot {\text{m}}^{\text{2}}\end{array}$

Conclusion:

The value is $I=0.61\text{kg}\cdot {\text{m}}^{\text{2}}$

(c)

To determine

To find: The axis about which it is harder to accelerate.

Explanation of Solution

Given information:

  $\begin{array}{l}m=1.8\text{kg}\\ \text{M}=\text{3}\text{.1}\text{}\text{kg}\end{array}$

The objects are wired together by very light, rigid pieces of wire.

Calculation:

Because of larger I value; it is harder to accelerate the array about the vertical axis.

Conclusion:

It is harder to accelerate the array.