A closed loop is made of a metal pipe and is rolling between two fixed pulleys as shown in the image on the left. What would be the path traced by point P when the loop completes one turn?
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For path-tracing (locus) problems, carefully define the moving point and the reference frame. Sometimes, the question might be simpler than a complex kinematic analysis suggests. Consider if the path traced is simply the shape of the track or guide itself.
Step 1: Understanding the Concept:
The question asks for the trajectory (path) of a specific point P on a closed loop as the loop rolls without slipping around two pulleys. This type of curve is known as a roulette curve, specifically related to a cycloid. The motion is a combination of translation (the loop moving forward) and rotation (the loop moving around its own path).
Step 2: Key Formula or Approach:
We can analyze the motion of point P in segments, corresponding to the different parts of the track it travels on.
1. Motion along the straight top section: The loop is moving horizontally. Point P is on the outer edge. It will trace an inverted cycloid-like arc as it moves from the back pulley to the front pulley. It starts at the top, moves down and forward, and then back up to the top.
2. Motion around the front semi-circular section: The whole loop rotates around the center of the right pulley. Point P will trace part of an epicycloid or a simple circular arc depending on the reference frame. Relative to the center of the loop, it's just moving in a circle. Since the loop itself is moving, P traces a curved path.
3. Motion along the straight bottom section: Similar to the top, P will trace another cycloid-like arc.
4. Motion around the back semi-circular section: Similar to the front pulley.
Step 3: Detailed Explanation:
Let's trace the path starting from point P at the top-most position as shown.
Top Section (Rightward travel): As the loop moves to the right, point P, which starts at the top of its own circular cross-section, will effectively roll along an imaginary line above the loop. It moves downwards and forwards, then upwards and forwards, tracing a cusp when it touches the imaginary line. Since P is on the outer edge, it won't touch the central path, so it will trace a smooth inverted arc. However, the question asks for the path of P on the large loop itself as it moves. As the loop moves along the top straight path, P rotates around the semicircular end.
Let's re-interpret the motion: The entire loop rotates. P starts at the top. As the loop makes one full revolution, P will travel around the entire perimeter of the loop's path.
First Half-Turn: P moves from the top, around the right pulley, to the bottom. While the loop moves along the top straight section, P is actually traversing the right semi-circular end of the loop. This motion will be a semi-circular arc.
Second Half-Turn: P moves from the bottom, around the left pulley, back to the top. While the loop moves along the bottom straight section, P is traversing the left semi-circular end of the loop. This motion will also be a semi-circular arc.
The problem is that the loop itself is translating. The path of P in a fixed reference frame is a combination of this rotation and translation.
Let's consider the point P's movement from its highest point. As the loop moves forward, P travels over the right pulley. This will generate a sharp, pointed cusp at the top of its trajectory as its vertical velocity becomes zero. It then travels down and forward.
As P goes around the bottom pulley, it will trace a rounded, circular arc at the bottom of its trajectory.
The combination of rolling along a straight line (creating a cycloid cusp) and rolling around a circle (creating a more rounded turn) results in a specific shape. The path of a point on the circumference of a circle rolling on another circle is an epicycloid. The path while rolling on a straight line is a cycloid.
At the top and bottom where the loop is tangent to the pulleys, the point P is at the furthest point from the center of rotation (the pulley axle). The speed of P is highest here.
The turns at the top and bottom will be different. The turn at the top will be sharp (a cusp), while the turn at the bottom will be a smooth semi-circle.
Step 4: Final Answer:
The question asks for the path traced by point P as the entire loop completes one turn. The loop is guided by two fixed pulleys. The path of any point on the loop, such as P, will trace a shape in space that is identical to the shape of the loop's centerline. This shape consists of two straight sections connected by two semicircles. Option (D) correctly depicts this shape.
