Understanding what is the maximum angular velocity with which the turntable can spin without the coin sliding is a classic physics problem that beautifully illustrates the interplay between friction, centripetal force, and rotational motion. This seemingly simple scenario has practical implications, from understanding how objects behave on rotating platforms to the design of high-speed machinery. At its core, the problem revolves around the concept of static friction providing the necessary centripetal force to keep an object moving in a circular path. If the rotational speed exceeds a certain limit, this crucial frictional force becomes insufficient, leading to the object sliding off.
The Fundamental Principle: Static Friction Meets Centripetal Force
To determine what is the maximum angular velocity with which the turntable can spin without the coin sliding, we must first grasp the core physics involved. When a coin rests on a spinning turntable, it naturally wants to continue moving in a straight line due to inertia. However, because it’s constrained to move in a circle, a force must be acting on it, pulling it towards the center of rotation. This inward-directed force is known as the centripetal force.
In the case of a coin on a turntable, the static friction force between the coin and the turntable surface is what provides this essential centripetal force. Static friction is the force that opposes the tendency of motion between two surfaces that are in contact but not yet sliding relative to each other. As the turntable spins faster, the demand for centripetal force increases. The static friction force can only provide a certain maximum amount of force before the coin begins to slip. Once the required centripetal force exceeds this maximum static friction, the coin will slide outwards.
Therefore, the critical condition for the coin not to slide is that the static friction force must be equal to or greater than the centripetal force required to keep the coin in its circular path. At the point of maximum angular velocity, these two forces are precisely equal.
Key Factors Influencing the Maximum Angular Velocity
Several key factors directly influence what is the maximum angular velocity with which the turntable can spin without the coin sliding:
1. Coefficient of Static Friction (μs)
The coefficient of static friction (μs) is a dimensionless quantity that quantifies the amount of friction between two surfaces. It represents the ratio of the maximum static friction force to the normal force pressing the surfaces together.
- A higher coefficient of static friction means there is more “grip” between the coin and the turntable. This allows for a greater maximum static friction force, which in turn means the turntable can spin faster before the coin slides.
- Conversely, a lower coefficient of static friction (e.g., a very smooth or oily surface) will result in a lower maximum static friction force, causing the coin to slide at much lower angular velocities.
- Typical values for μs range from 0 (no friction) to greater than 1 (very high friction, like rubber on concrete).
2. Distance of the Coin from the Center of the Turntable (r)
The distance of the coin from the center of the turntable (r), also known as the radius of the circular path, plays a crucial role.
- The centripetal force required to keep an object moving in a circle is directly proportional to the radius of that circle. This means that the further the coin is from the center, the greater the centripetal force required to keep it moving in the circular path at a given angular velocity.
- Consequently, for a given coefficient of friction, a coin placed closer to the center will be able to withstand a higher angular velocity than a coin placed further out. This is why objects tend to fly off the edges of spinning objects first.
3. Acceleration Due to Gravity (g)
The acceleration due to gravity (g) is a constant value on Earth, approximately 9.81 m/s².
- Gravity is essential because it determines the normal force acting on the coin. The normal force is the force exerted by the surface perpendicular to the coin, and in this horizontal setup, it’s equal to the coin’s weight (mass × g).
- Since the maximum static friction force is directly proportional to the normal force (Ffriction = μs × Fnormal), gravity indirectly influences the maximum available friction. While ‘g’ is constant for most terrestrial experiments, it’s a fundamental component of the normal force calculation.
The Derivation of the Formula
To precisely calculate what is the maximum angular velocity with which the turntable can spin without the coin sliding, we can derive a formula based on the equilibrium condition where static friction provides the centripetal force.
1. Centripetal Force (Fc): The force required to keep an object of mass ‘m’ moving in a circle of radius ‘r’ with angular velocity ‘ω’ is given by:
$F_c = m \cdot r \cdot \omega^2$
2. Maximum Static Friction Force (Ff_max): The maximum static friction force that can be exerted on the coin is given by:
$F{f\max} = \mus \cdot F{normal}$
Since the turntable is horizontal, the normal force ($F_{normal}$) acting on the coin is equal to its weight, which is mass (m) times the acceleration due to gravity (g):
$F_{normal} = m \cdot g$
So, the maximum static friction force is:
$F{f\max} = \mu_s \cdot m \cdot g$
3. Equating Forces for Maximum Velocity: At the maximum angular velocity ($\omega_{max}$), the required centripetal force is precisely equal to the maximum static friction force:
$Fc = F{f\_max}$
$m \cdot r \cdot \omega{max}^2 = \mus \cdot m \cdot g$
4. Solving for $\omega_{max}$: Notice that the mass ‘m’ of the coin cancels out from both sides of the equation. This is a crucial insight: the mass of the coin does not affect the maximum angular velocity.
$r \cdot \omega{max}^2 = \mus \cdot g$
$\omega{max}^2 = \frac{\mus \cdot g}{r}$
$\omega{max} = \sqrt{\frac{\mus \cdot g}{r}}$
This formula, $\omega{max} = \sqrt{\frac{\mus \cdot g}{r}}$, is the definitive answer to what is the maximum angular velocity with which the turntable can spin without the coin sliding. The angular velocity ($\omega$) is typically measured in radians per second (rad/s).
How to Apply the Formula: A Step-by-Step Guide
To apply the formula and calculate what is the maximum angular velocity with which the turntable can spin without the coin sliding, follow these steps:
- Identify the Values:
- Coefficient of static friction ($\mu_s$): This value must be determined experimentally or looked up for the specific materials (coin and turntable surface). It is a dimensionless quantity.
- Distance of the coin from the center of the turntable (r): This is the radius of the circular path. Ensure it is measured in meters (m). If given in centimeters, convert it (1 cm = 0.01 m).
- Acceleration due to gravity (g): For calculations on Earth, use $g \approx 9.81 \, m/s^2$.
2. Substitute the Values into the Formula:
Plug the identified values for $\mu_s$, g, and r into the equation:
$\omega{max} = \sqrt{\frac{\mus \cdot g}{r}}$
- Calculate the Result:
Perform the arithmetic operations carefully.
- Multiply $\mu_s$ by g in the numerator.
- Divide the result by r.
- Take the square root of the final value.
The resulting $\omega_{max}$ will be in radians per second (rad/s).
Example Calculation
Let’s use the example provided in the AI overview content to illustrate the application:
- Coefficient of static friction ($\mu_s$) = 0.85
- Distance of the coin from the center (r) = 0.19 meters
- Acceleration due to gravity (g) = 9.81 m/s²
Now, substitute these values into the formula:
$\omega_{max} = \sqrt{\frac{0.85 \cdot 9.81}{0.19}}$
First, calculate the numerator:
$0.85 \cdot 9.81 = 8.3385$
Next, divide by the radius:
$\frac{8.3385}{0.19} \approx 43.8868$
Finally, take the square root:
$\omega_{max} = \sqrt{43.8868} \approx 6.62 \, rad/s$
(Note: There’s a slight discrepancy in the example’s result (6.73 rad/s vs 6.62 rad/s). This often arises from rounding differences during intermediate steps or using a slightly different value for ‘g’. The principle remains the same.)
Another example given:
- Coin mass = 4.20 g (irrelevant, as mass cancels out)
- Distance from center (r) = 15.0 cm = 0.15 m
- Coefficient of static friction ($\mu_s$) = 0.700
- Acceleration due to gravity (g) = 9.81 m/s²
Applying the formula:
$\omega_{max} = \sqrt{\frac{0.700 \cdot 9.81}{0.15}}$
$\omega_{max} = \sqrt{\frac{6.867}{0.15}}$
$\omega_{max} = \sqrt{45.78}$
$\omega_{max} \approx 6.77 \, rad/s$
This calculation confirms that for specific values of friction and radius, the maximum angular velocity can be precisely determined.
Converting Angular Velocity to More Familiar Units
While radians per second (rad/s) is the standard unit for angular velocity in physics, it’s often more intuitive to think in terms of revolutions per minute (RPM) or revolutions per second (RPS).
- From Radians per Second to Revolutions per Second (RPS):
There are $2\pi$ radians in one complete revolution.
$RPS = \frac{\omega_{max}}{2\pi}$
- From Radians per Second to Revolutions per Minute (RPM):
First, convert to RPS, then multiply by 60 seconds per minute.
$RPM = \frac{\omega_{max}}{2\pi} \cdot 60$
Using our first example where $\omega_{max} \approx 6.62 \, rad/s$:
$RPS = \frac{6.62}{2\pi} \approx \frac{6.62}{6.283} \approx 1.05 \, RPS$
$RPM = 1.05 \cdot 60 \approx 63 \, RPM$
So, in this scenario, the turntable could spin at approximately 63 revolutions per minute before the coin would begin to slide.
Practical Considerations and Limitations
While the formula provides a precise theoretical answer to what is the maximum angular velocity with which the turntable can spin without the coin sliding, several practical considerations and limitations should be kept in mind:
- Homogeneity of the Turntable Surface: The formula assumes a uniform coefficient of static friction across the entire turntable surface. In reality, surfaces can have irregularities or varying degrees of cleanliness that affect friction.
- Coin Condition: The type of coin (material, wear, cleanliness) can influence its coefficient of static friction. A greasy coin will have a lower $\mu_s$ than a clean one.
- Air Resistance: At very high angular velocities, air resistance on the coin could become a factor, though for typical turntable speeds, its effect is usually negligible compared to friction.
- Turntable Flatness and Levelness: The formula assumes a perfectly flat and level turntable. Any tilt would introduce a gravitational component that would affect the normal force and thus the available static friction.
- Dynamic vs. Static Friction: The formula uses the coefficient of static friction. Once the coin starts to slide, the friction transitions to kinetic friction, which is generally lower than static friction. This means that once it starts to slide, it will accelerate outwards even more rapidly.
- Measurement Accuracy: The accuracy of the calculated $\omega{max}$ depends on the accuracy of the measured values for $\mus$ and r. Determining $\mu_s$ precisely can be challenging.
Conclusion
The question of what is the maximum angular velocity with which the turntable can spin without the coin sliding is a perfect example of applied physics that combines concepts of forces, motion, and friction. By understanding that the static friction force must provide the necessary centripetal force, we can derive a simple yet powerful formula: $\omega{max} = \sqrt{\frac{\mus \cdot g}{r}}$.
This formula highlights that the maximum angular velocity is directly proportional to the square root of the coefficient of static friction and inversely proportional to the square root of the distance from the center. Counter-intuitively, the mass of the coin itself does not affect this maximum velocity. This understanding is not only fundamental to physics education but also offers insights into how objects behave in rotational systems, from amusement park rides to industrial centrifuges.
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