Question

Suppose you are riding on a roller coaster, which moves through a vertical circular loop. Show that your apparent weight at the bottom of the loop is six times your weight when you experience weightlessness at the top, independent of the size of the loop. Assume that friction is negligible.

Short answer

Question: At the bottom of a roller coaster loop, the apparent weight of a person is six times their weight when they experience weightlessness at the top of the loop. Demonstrate the validity of this statement. Answer: The apparent weight at the bottom of a roller coaster loop is six times the person's weight when they experience weightlessness at the top, independent of the size of the loop and assuming negligible friction. This is shown through the analysis of centripetal force, gravitational force, apparent weight, and conservation of energy for the roller coaster at the top and bottom of the loop.

Step-by-step solution

Define the variables and equations

Let's denote the mass of the person as m, the speed of the roller coaster at the top and bottom as v_top and v_bottom, the radius of the loop as R, the gravitational acceleration as g, and the normal force (apparent weight) as N. We know that the centripetal force required for the person to stay in circular motion is given by F_c = m*v^2/R, where v is the velocity at the given point on the loop.

Determine the conditions for weightlessness

At the top of the loop, the person will experience weightlessness when the gravitational force (m*g) acting on them equals the centripetal force acting upwards. At this point, the apparent weight (normal force N) will be zero. The condition for this to happen is: m*g = m*v_top^2/R

Determine the velocity at the bottom of the loop

At the top of the loop, the roller coaster has a potential energy (m*g*h) and kinetic energy (1/2*m*v_top^2). When it reaches the bottom, all the potential energy will be converted into kinetic energy. Since the height of the roller coaster from the top to the bottom of the loop is 2R, we can write the conservation of energy equation as: m*g*2R + 1/2*m*v_top^2 = 1/2*m*v_bottom^2

Calculate the normal force at the bottom

At the bottom of the loop, there are two forces acting on the person: the gravitational force (downwards) and the centripetal force (upwards). We can write the total force equation as: N - m*g = m*v_bottom^2/R

Find the ratio between the apparent weight at the bottom and the top

We need to show that the apparent weight (normal force) at the bottom of the loop is six times the weight of the person when they experience weightlessness at the top. First, we can rearrange the equation for weightlessness (Step 2): v_top^2 = R*g Now substitute this expression into the conservation of energy equation (Step 3) and solve for v_bottom^2: m*g*2R + 1/2*m*R*g = 1/2*m*v_bottom^2 Rearranging and solving for v_bottom^2 gives: v_bottom^2 = 5*R*g Finally, substitute this expression for v_bottom^2 into the total force equation (Step 4) and solve for the normal force (apparent weight) at the bottom: N - m*g = m*(5*R*g)/R N = m*g*5 From this equation, we can see that the apparent weight at the bottom is six times the weight of the person at the top, when they experience weightlessness, independently of the size of the loop. Therefore, we have shown that the apparent weight at the bottom of the loop is six times the weight of the person when they experience weightlessness at the top, independent of the size of the loop, and assuming that friction is negligible.

Key concepts

Apparent Weight in Circular Motion

When you're experiencing the thrills of a rollicking roller coaster ride, your body is subjected to various forces that can make you feel heavier or lighter than usual. This perceived change in weight is known as 'apparent weight.' During a vertical circular motion, such as when a roller coaster zooms through a loop, your apparent weight can vary dramatically.

At the bottom of the loop, the combination of gravitational pull and the need for a centripetal force to keep you in circular motion results in a greater normal force exerted by the coaster's seat on your body. This is why you feel squashed or heavier. In contrast, at the top of the loop, if the centripetal force needed for circular motion is just supplied by gravity, you feel weightless because the seat no longer needs to push against you. This nuance in physics not only explains the varying sensations of weight but also plays a crucial role in designing safe and exciting roller coaster rides.

Weightlessness in Physics

Ever wonder what astronauts feel when they're floating in space? This sensation is called 'weightlessness,' and it's not just for spacefarers—roller coaster enthusiasts can experience it too! At the peak of a roller coaster loop, if the force of gravity is perfectly balanced by the centripetal force, you would feel weightless. That's because the normal force—the force that typically supports you when you're standing on solid ground—is zero.

This doesn't mean gravity has stopped working; it's just that gravity alone is providing the necessary centripetal force to navigate the top of the loop, leaving no additional force to act on your body. Hence, you temporarily experience a state of 'free fall' while being safely contained within the coaster, giving you that heart-stopping moment of weightlessness before gravity yanks you back into your seat.

Conservation of Energy in Roller Coasters

Roller coasters are a dynamic classroom for understanding the conservation of energy—a fundamental principle in physics. As the coaster climbs and descends, it continually converts potential energy (due to its height) into kinetic energy (due to its motion) and vice versa. The sum of kinetic and potential energy at any point is constant, assuming no energy is lost to friction or air resistance.

When a roller coaster reaches the apex of its track, it has maximum potential energy, which then transforms into kinetic energy as it races down. That kinetic energy is what propels the coaster through loops and over hills. By calculating the energy at the top and bottom of a loop, we can determine the necessary speed for the coaster to safely complete the loop, and by extension, the forces riders will experience. This manipulation and preservation of energy are what allow for the design of exhilarating yet secure roller coaster experiences.

Centripetal Force Calculations

Centripetal force is the invisible hand that keeps objects moving in a circular path, pulling them toward the center of rotation. When a roller coaster car speeds through a loop, it's the centripetal force that prevents it from flying off the track. Calculating this force involves knowing the mass of the object, the speed at which it's traveling through the curve, and the radius of the curve itself.

The equation for centripetal force (\( F_c = \frac{m \times v^2}{R} \) where m is mass, v is velocity, and R is radius) allows engineers to design loops that are both thrilling and safe. Riders' apparent weight and the sensations they experience, such as weightlessness, are tied directly to the balance of centripetal force and gravity, showcasing the elegant dance between the forces at play in circular motion.