Problem 51 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A coin is placed \(12.0 \mathrm{~cm}\) from the axis of a rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 35.0 rpm (revolutions per minute) is reached, at which point the coin slides off. What is the coefficient of static friction between the coin and the turntable?

Step-by-Step Solution

Answer

The coefficient of static friction is approximately 0.14.

Step 1: Understand the forces

Identify the relevant forces acting on the coin. The frictional force is what keeps the coin from sliding off as the turntable rotates, and this frictional force equals the centripetal force required to keep the coin moving in a circle.

Step 2: Establish the relationship

The maximum static frictional force before sliding occurs is given by:\[ f_s = \mu_s \cdot N \] where \( f_s \) is the static frictional force, \( \mu_s \) is the coefficient of static friction, and \( N \) is the normal force. Because the turntable is horizontal, \( N = mg \), where \( m \) is the mass of the coin and \( g \) is the acceleration due to gravity.

Step 3: Equate with centripetal force

The centripetal force needed to keep the coin moving in a circle is given by:\[ f_c = \frac{mv^2}{r} \] where \( v \) is the linear velocity and \( r \) is the radius of the circle (12 cm or 0.12 m). This force is provided by static friction until the maximum static friction is exceeded.

Step 4: Use the relationship between linear and angular velocity

The linear velocity \( v \) can be expressed in terms of angular velocity \( \omega \): \[ v = r\omega \]. Since the angular speed is given in revolutions per minute (rpm), convert it to radians per second: \[ \omega = 35 \times \frac{2\pi}{60} \]

Step 5: Solve for the velocity

Substitute into the linear velocity formula:\[ v = 0.12 \cdot 35 \times \frac{2\pi}{60} \] Calculate \( v \) to find the linear velocity of the coin.

Step 6: Calculate the centripetal force

Using the value of \( v \) found in the previous step, calculate the centripetal force \( f_c = \frac{mv^2}{r} \) needed for the coin to move in the circular path.

Step 7: Equate frictional and centripetal forces

Equate the maximum static frictional force and the calculated centripetal force: \[ \mu_s mg = \frac{mv^2}{r} \] Simplify to solve for the coefficient of static friction \( \mu_s \): \[ \mu_s = \frac{v^2}{rg} \]

Step 8: Calculate coefficient of static friction

Substitute \( v \) and \( r = 0.12 \) m, and \( g = 9.81 \) m/s² into the equation to find \( \mu_s \).

Key Concepts

Centripetal Force

Centripetal force is a crucial concept when dealing with objects moving in circular paths. It refers to the force that keeps an object moving in a circular direction. Think of it as the "center-seeking" force because it acts towards the center of the circle.
When the coin is placed on a rotating turntable, it needs this centripetal force to keep moving in a circle. Without this force, the coin would simply move off in a straight line due to its inertia.
The centripetal force is not a separate kind of force. It's provided by other forces such as gravity, tension, or in this case, friction. For our coin on the turntable, static friction acts as the centripetal force.

• Formula: \[ f_c = \frac{mv^2}{r} \]
• Here, \( m \) is the mass of the object, \( v \) is the linear velocity, and \( r \) is the radius of the circle.
• In simpler terms, the faster the turntable spins, or the further the coin is from the center, the more force is required to keep the coin moving in its circle.

Linear Velocity

Linear velocity is a measure of how fast something moves along a path. For a coin on a rotating turntable, it's about how fast the coin travels along its circular path. You may have heard of it as "speed," but linear velocity includes both speed and direction.
To find the linear velocity \( v \) of the coin, we need to relate it to the angular velocity \( \omega \). For circular movement, the linear velocity is tied to the distance from the center, which is known as the radius \( r \).

• Formula: \[ v = r \omega \]
• Where \( \omega \) is the angular velocity (how fast the entire circle is completed), and \( r \) is the radius (distance from the axis of rotation to the coin).
• This shows that if a point is further out from the center of rotation, it will have a higher linear velocity.

This understanding helps us calculate the linear velocity which is then used to find the centripetal force.

Angular Velocity

Angular velocity is all about how quickly an object rotates around a central point. In the case of the coin on the turntable, we're looking at how fast the turntable spins.
It's usually given in revolutions per minute (rpm), but for calculations, we convert it to radians per second. This is because one full circle is \( 2\pi \) radians.

• Conversion formula: \[ \omega = \text{{rpm}} \times \frac{2\pi}{60} \]
• Here, we multiply the \( \text{{rpm}} \) by \( 2\pi \) to change revolutions into radians, and then divide by 60 to convert minutes into seconds.
• Understanding \( \omega \) helps us calculate linear velocity, and thus find how forces like friction are affected by rotational speed.

Comprehending how angular velocity works is key to understanding many real-world scenarios involving rotation, like why a car needs different tire grip on a slippery road to maintain speed and direction.