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.

Key concepts

Lorentz Contraction

Lorentz Contraction is a fascinating phenomenon of special relativity that occurs when an object moves at a significant fraction of the speed of light relative to an observer. Due to this effect, the length of the object is perceived to be shorter along the direction of motion. Let's unravel this further.

• The contraction only affects dimensions parallel to the velocity.
• The formula to calculate contracted length is \(L' = \frac{L}{\gamma(v)}\), where \(L\) is the proper length and \(\gamma(v)\) is the Lorentz factor.
• The Lorentz factor, \(\gamma(v)\), is defined as \(\gamma(v) = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), where \(c\) is the speed of light.

In our problem, an 18-foot pole moving with a high velocity appears to be only 6 feet long in a different frame due to Lorentz contraction. Notice how the pole's observed length decreases as the speed increases.

Reference Frames

In physics, a reference frame is a point of view from which observations are made. Both motion and measurements can differ depending on the reference frame you choose. Reference frames can be stationary or moving, which is crucial in relativity.

• There are often two main frames: the stationary frame \(S\) and the moving frame \(S'\).
• In this exercise, frame \(S\) is where the hole is at rest, while \(S'\) moves along with the pole.
• Mathematical transformations are used to convert observations from one frame to another.

From frame \(S'\), the movement of the pole is such that it experiences length contraction, allowing it to pass through the hole in the plate satisfactorily. Though the hole stays the same, the pole's perception changes as it switches frames.

Relativistic Velocity

Relativistic velocity comes into play when you deal with objects moving at great speeds, often close to the speed of light. As velocities approach the speed of light, classical mechanics gives way to relativistic mechanics to accurately describe motion.

• Relativistic velocity affects length contraction, affecting how we measure speeds and distances.
• An important feature is that time, space, and velocity are interlinked, leading to relativistic effects.
• The problem describes the pole moving with velocity \((v, -w, 0)\), showing the velocity is not just in one straight line but has components.

Navigating through these velocities, observers in different frames witness different phenomena due to these relativistic principles. This fascinating interplay makes sure that the 18-foot pole, when examined from frame \(S'\), comfortably shrinks to 6 feet, showcasing Special Relativity in action!