Problem 63 - College Physics

Question

A steel ball rolls down an incline into a loop-theloop of radius \(R\) (v Fig. 8.52a). (a) What minimum speed must the ball have at the top of the loop in order to stay on the track? (b) At what vertical height \((h)\) on the incline, in terms of the radius of the loop, must the ball be released in order for it to have the required minimum speed at the top of the loop? (Neglect frictional losses.) (c) Figure \(8.52 \mathrm{~b}\) shows the loop-the-loop of a roller coaster. What are the sensations of the riders if the roller coaster has the minimum speed or a greater speed at the top of the loop? [Hint: In case the speed is below the minimum, seat and shoulder straps hold the riders in.

Step-by-Step Solution

Answer

(a) Minimum speed is \( v = \sqrt{gR} \). (b) Height is \( h = \frac{5}{2}R \). (c) At minimum speed, riders feel weightless.

Step 1: Analyze the Forces at the Top of the Loop

At the top of the loop, for the ball to stay on track, the gravitational force must provide the centripetal force required to maintain the circular motion. The forces acting on the ball are its weight, \( mg \), and the normal force from the track. For minimum speed, at the threshold of losing contact, the normal force becomes zero. Thus, the centripetal force is solely provided by gravity: \( \frac{mv^2}{R} = mg \).

Step 2: Solve for Minimum Speed at the Top of the Loop

To find the minimum speed \( v \) of the ball at the top, we solve the equation from Step 1: \( \frac{mv^2}{R} = mg \). Simplifying gives \( v^2 = gR \) and so \( v = \sqrt{gR} \).

Step 3: Use Energy Conservation to Find Initial Height

Initially, the ball is at height \( h \) with potential energy, and at the top of the loop, it has both potential and kinetic energy. Using the conservation of energy, the potential energy at the beginning, \( mgh \), equals the total energy at the top of the loop: \( mg(2R) + \frac{1}{2}mv^2 \). Plug \( v = \sqrt{gR} \) into the energy equation: \( mgh = mg(2R) + \frac{1}{2} m(gR) \). Solve for \( h \): \( h = \frac{5}{2}R \).

Step 4: Discuss the Sensations of Riders

If the ball maintains exactly the minimum speed at the top, riders would feel weightless as the normal force is zero. Above minimum speed, riders still experience downward acceleration but also a non-zero normal force, which means they would feel pushed into their seats.

Key Concepts

Centripetal Force

Centripetal force is essential to keep an object moving in a circular path. In the context of the loop-the-loop problem, it acts as the force that holds the ball on the track as it moves around the loop. A crucial aspect to understand is how this force operates when the ball is at the top of the loop. At this highest point, gravity contributes to the centripetal force needed for circular motion.

• When an object moves in a circle, centripetal force is directed towards the center of the circle.
• It can be calculated using the formula: \( F_c = \frac{mv^2}{R} \).
• For this exercise, gravity must provide enough force to act as the centripetal force when at the top of the loop.

At the minimum speed, the role of the normal force vanishes, and gravity must account fully for the centripetal force to keep the ball on track. This is why we set the centripetal force equal to the gravitational force at the top to find the minimum speed.

Motion in a Vertical Circle

Motion in a vertical circle, like in roller coasters or loop-the-loops, involves a dynamic exchange between potential and kinetic energy.

• The energy states vary as the ball or object moves through points of different heights in the circle.
• Gravity's constant presence affects speed and the apparent weight an observer might feel.

When objects move through a vertical loop, they begin with more energy than what is just required to reach the top. The extra energy ensures they maintain necessary speed to avoid falling off. For this reason, ideal looping starts with downward motion where gravitational energy converts to kinetic energy, making certain the object completes the loop safely. Riders feel various sensations, ranging from weightless moments at the top to being pressed into their seats below, depending on speed variations.

Conservation of Energy

The principle of conservation of energy is pivotal in understanding how energy transitions in motion, especially in systems like loops.

• Energy is not lost; it is transformed from one form to another, typically from potential energy (due to height) to kinetic energy (due to motion) and vice versa.
• In frictionless scenarios, the total mechanical energy is conserved.

For a ball released at a height to complete a vertical loop, conservation of energy dictates that the ball will have sufficient speed at the loop's top. The energy equation applies, \( mgh = mg(2R) + \frac{1}{2}mv^2 \), indicating potential energy at the start equates to both potential and kinetic energy at the loop's top. By calculating, it suggests that initial release height must be at least \( h = \frac{5}{2}R \), ensuring the ball has enough energy to maintain motion throughout the loop.