Question

An 18-foot pole, while remaining parallel to the \(x\)-axis, moves with velocity \((v,-w, 0)\) relative to frame \(S\), where \(\gamma(v)=3\), and \(w\) is positive. The centre of the pole passes the centre of a 9 -foot hole in a plate that coincides with the plane \(y=0\). Explain, from the point of view of the usual second frame \(S^{\prime}\) moving with velocity \(v\) relative to \(S\), how the pole gets through the hole.

Short answer

Explain your answer. Answer: Yes, in frame \(S'\), the pole can pass through the 9-foot hole because its length is contracted to 6 feet, which is less than the hole's length of 9 feet.

Step-by-step solution

Calculate the length of the pole in frame \(S'\)

Using the Lorentz factor and the length contraction formula, we can calculate the length of the pole in frame \(S'\). The formula is given by \(L' = L/\gamma(v)\), where \(L'\) is the length in frame \(S'\), \(L\) is the length in frame \(S\), and \(\gamma(v)\) is the Lorentz factor. We are given \(L = 18\) feet and \(\gamma(v) = 3\). Plugging in these values, we get: \(L' = \frac{18}{3} = 6\) feet So, the length of the pole in frame \(S'\) is 6 feet.

Calculate the length of the hole in frame \(S'\)

Since the pole is moving in the x-direction and the hole is on the y=0 plane, there is no length contraction for the hole in the \(S'\) frame. Thus, the length of the hole is the same in both frames. Hole length in frame \(S'\): \(L_h' = 9\) feet.

Assess if the pole can pass through the hole in frame \(S'\)

We have found the lengths of the pole and the hole in frame \(S'\). The pole can pass through the hole in frame \(S'\) if its length is less than or equal to the hole's length. Pole length in frame \(S'\): \(L' = 6\) feet Hole length in frame \(S'\): \(L_h' = 9\) feet Since \(L' <= L_h'\) (6 feet <= 9 feet), the pole in frame \(S'\) can indeed pass through the hole. In conclusion, from the point of view of frame \(S'\), the pole gets through the hole because its length is contracted to 6 feet, which is less than the hole's length of 9 feet.