Question

You put three identical coins on a turntable at different distances from the center and then turn the motor on. As the turntable speeds up, the outermost coin slides off first, followed by the one at the middle distance, and, finally, when the turntable is going the fastest, the innermost one. Why is this? a) For greater distances from the center, the centripetal acceleration is higher, and so the force of friction becomes unable to hold the coin in place. b) The weight of the coin causes the turntable to flex downward, so the coin nearest the edge falls off first. c) Because of the way the turntable is made, the coefficient of static friction decreases with distance from the center. d) For smaller distances from the center, the centripetal acceleration is higher.

Short answer

Answer: For greater distances from the center, the centripetal acceleration is higher, and so the force of friction becomes unable to hold the coin in place.

Step-by-step solution

Understand Centripetal Acceleration

Centripetal acceleration is the acceleration of an object moving in a circle at a constant speed. It acts always towards the center of the circle, along the radius. The centripetal acceleration is given by the equation a_c = r * ω^2, where a_c is the centripetal acceleration, r is the distance from the center, and ω is the angular velocity.

Investigate the Options

We need to understand which factor is causing the coins to slide off the turntable at different distances from the center, considering the role of centripetal acceleration, centripetal force, and friction. Let's analyze each of the given explanations one by one: a) For greater distances from the center, the centripetal acceleration is higher, and so the force of friction becomes unable to hold the coin in place. b) The weight of the coin causes the turntable to flex downward, so the coin nearest the edge falls off first. c) Because of the way the turntable is made, the coefficient of static friction decreases with distance from the center. d) For smaller distances from the center, the centripetal acceleration is higher.

Determine the Role of Centripetal Acceleration

We already know that centripetal acceleration is given by the equation a_c = r * ω^2. This equation shows that as the distance from the center (r) increases, the centripetal acceleration also increases. Therefore, option a) is consistent with this, but option d) contradicts it.

Consider the Role of Friction

Friction is the force resisting relative motion between two surfaces in contact. In this case, it is the force that holds the coins on the turntable. The force of static friction (f_s) is given by f_s = μ_s * N, where μ_s is the coefficient of static friction and N is the normal force (typically equal to the weight of the object for horizontal surfaces). While the friction is vital in this context, there's no reason to believe that the coefficient of static friction (μ_s) will change with distance from the center as suggested in option c).

Eliminate Incorrect Options

Based on our understanding of the centripetal acceleration, we can eliminate the incorrect options: c) The coefficient of static friction doesn't change with distance from the center unless specifically stated. d) The centripetal acceleration is higher for greater distances, not smaller distances.

Choose the Correct Answer

Considering the role of centripetal acceleration and friction, option a) seems to be the best explanation: For greater distances from the center, the centripetal acceleration is higher, and so the force of friction becomes unable to hold the coin in place. The weight of the coin causing the turntable to flex downward (option b) might play a small role, but it is not the primary reason for the coins sliding off at different distances from the center.

Key concepts

Friction

Imagine the feeling of your hand gliding along a smooth table versus pressing it against the table and sliding it with force. That resistance you feel is friction, an essential concept in understanding the behavior of objects in motion. Friction is the force that resists the relative motion between two surfaces in contact. It doesn't just prevent things from sliding; it also plays a pivotal role in enabling motion, as is the case when we walk without slipping.

In the physical world, friction serves as the gatekeeper for motion, and it manifests in two primary forms: static friction and kinetic friction. Static friction acts on objects that are not in motion relative to each other, while kinetic friction takes over once the objects start to slide. The static friction force is what anchors the coins to the turntable. It is powerful enough to maintain the coin's position, but only until a certain threshold is surpassed. When the centripetal force necessary to keep the coin circling exceeds the maximum static friction force, the coin escapes its circular path and slides off the turntable.

Coefficient of Static Friction

To appreciate the unsung hero in the friction saga, let's focus on the coefficient of static friction, symbolized as \( \mu_s \). Picture this as a measure of 'stickiness' between two surfaces when no movement is occurring. It varies based on the materials in contact - rubber on concrete is stickier than ice on steel, for instance. The coefficient of static friction contributes to calculating the maximum static frictional force using the equation \( f_s = \mu_s \cdot N \) where \( N \) is the normal force and \( f_s \) is the static frictional force.

This coefficient doesn't just sit idly; it determines how much force it takes to start sliding an object. In the turntable puzzle, the higher the \( \mu_s \) of the coin's surface, the harder it will be to dislodge it from its rhythmic spinning. However, in the absence of unique materials or structural differences on the turntable, there's little reason to assume that the \( \mu_s \) changes with distance from the center. It's a constant companion to the coins, right up until the point when static gives way to kinetic friction and the coins embark on their ill-fated slide off the edge.

Angular Velocity

Now, let's delve into angular velocity, an elegant dance of motion in a circle. Represented by the Greek letter \( \omega \), angular velocity measures how fast an object rotates or revolves relative to another point, namely the center of rotation. Just as linear velocity defines the speed in a straight line, angular velocity is all about the speed in a curve. For example, figure skaters pull their arms in to spin faster - this is them manipulating their moment of inertia to change their angular velocity without an external torque.

In the context of our turntable and its revolting coins, angular velocity is the variable that dictates how swiftly the platform spins. It's intricately linked to centripetal acceleration, as shown by the formula \( a_c = r \cdot \omega^2 \). A juicy fact is that centripetal acceleration ramps up with both the distance from the center (\( r \) ) and the square of angular velocity (\( \omega^2 \) ). This means that as we get further from the center, or as our turntable zips faster, the centripetal force — directed towards the center and crucial to keeping the coin circling — grows. If the centripetal force outstrips the static friction force, which hinges on \( \mu_s \) and \( N \) , and the coins aren't sticky enough, it results in the outermost coins succumbing to a spiraling fate, sliding outward as they lose their grip on the spinning world beneath them.