Chapter 26, Problem 65GP

To determine

To Show: The boy can fit the pole within the barn and when he is running the barn look even shorter.

Explanation of Solution

Given data:

Length of pole, ${\mathcal{l}}_0=15\text{m}$

Length of barn, $\mathcal{l}=12\text{m}$

Formula used:

The length of an object moving relative to an observer is measured to be shorter along its direction of motion than when it is at rest.

Length contraction

  $\mathcal{l}={\mathcal{l}}_0\sqrt{1-v^2/c^2}$

Calculation:

When the boy run toward the farm. In the boy’s frame of reference, the pole remains at rest and its length will remain 15.0 m. But in the barn’s frame of reference the length of pole would contract in size, and it possibly fit into the barn. The length of barn is 12.0m. The boy should move with such a speed that the contracted length of pole become 12.0 m and it would fit within barn.

The original length of pole is ${\mathcal{l}}_0=15.0\text{m}$ and the contracted length of pole is $\mathcal{l}=12.0\text{m}$

  $\mathcal{l}={\mathcal{l}}_0\sqrt{1-v^2/c^2}$

  $\begin{array}{c}v=c\sqrt{1-{\mathcal{l}}^2/{\mathcal{l}}_0^2}\\ =c\sqrt{1-{(12.0\text{m})}^2/{(15.0\text{m})}^2}\\ =0.6c\end{array}$

In the boy’s frame of reference, the barn is moving and thus the length of the barn as he would measure it is less than the length of the pole.

  $L=L_o\sqrt{1-{\left(\frac{v}{c}\right)}^2}$

  $L=(12.0\text{m})\sqrt{1-{(0.60)}^2}$

  $L=9.6\text{m}$

As the boy runs toward the barn, relativity requires that the (relatively moving) barn contract in size, making the barn even shorter than its rest length of 12.0 m. Thus it is impossible, in the boy’s frame of reference, for the barn to be longer than the pole.

Conclusion:

In the boy’s frame of reference, it is impossible to fit a pole within the barn.