A coin is placed 33 cm from the center of a
horizontal turntable, initially at rest. The
turntable then begins to rotate. When the
speed of the coin is 100 cm/s (rotating at a
constant rate), the coin just begins to slip.
The acceleration of gravity is 980 cm/s2 .
What is the coefficient of static friction be-
tween the coin and the turntable?
Well, well, well! Looks like we've got a slippery situation here, quite literally! Let's get to the bottom of it, shall we?
First, we need to find the acceleration of the coin. Since the coin is about to slip, the force of static friction is at its maximum, balancing out the centrifugal force.
The centrifugal force can be calculated using F = mv²/r, where m is the mass of the coin and v is its velocity. So, the centrifugal force is given by F = mv²/r.
Now, the force of static friction can be calculated using F = μN, where μ is the coefficient of static friction and N is the normal force. In this case, the normal force is equal to the weight of the coin, which is mg, where g is the acceleration due to gravity.
Since the coin is at rest on the turntable, the force of static friction is equal to the centripetal force. So, we have μN = mv²/r.
If we substitute the value of N as mg and rearrange the equation, we get μmg = mv²/r.
Now, let's substitute the given values. The radius, r, is 33 cm or 0.33 m. The speed, v, is 100 cm/s or 1 m/s. And the acceleration due to gravity, g, is 980 cm/s² or 9.8 m/s².
Plugging these values into the equation, we have μmg = mv²/r
μ(0.33)(mg) = (m)(1²)/(0.33)
The mass of the coin cancels out on both sides of the equation, leaving us with:
μ(0.33)(g) = 1/(0.33)
Now, we just need to solve for μ. *drumroll*
μ = 1/(0.33)(0.33)(g)
And now, let the calculations commence!
To find the coefficient of static friction between the coin and the turntable, we can use the concept of centripetal force.
The centripetal force acting on the coin is provided by the static friction between the coin and the turntable. At the point where the coin just begins to slip, the static friction is at its maximum value.
The formula for centripetal force (Fc) is given by:
Fc = m * a
Where:
m is the mass of the coin
a is the centripetal acceleration
First, let's find the mass of the coin. We know that the weight of an object is given by:
Fg = m * g
Where:
Fg is the weight of the coin
g is the acceleration due to gravity
In this case, the weight of the coin provides the normal force between the coin and the turntable. The normal force cancels out the force of gravity.
Fg = m * g
Now, let's find the mass of the coin:
m = Fg / g
Next, let's find the centripetal acceleration.
The formula for centripetal acceleration (ac) is:
ac = ω^2 * r
Where:
ω is the angular velocity of the coin
r is the radius of the turntable
We know that the speed of the coin (v) is related to the angular velocity by:
v = ω * r
Rearranging the equation, we can solve for ω:
ω = v / r
Substituting this back into the formula for centripetal acceleration:
ac = (v / r)^2 * r
Now, let's calculate the centripetal acceleration.
Next, we can substitute the values for mass (m) and centripetal acceleration (ac) into the formula for centripetal force:
Fc = m * ac
Finally, the maximum value of static friction (Fs) is equal to the centripetal force (Fc) at the point where the coin just begins to slip.
Fs = Fc
The maximum value of static friction is given by:
Fs = μs * N
Where:
μs is the coefficient of static friction
N is the normal force between the coin and the turntable
Since the weight of the coin provides the normal force, we can substitute the formula for weight (Fg) into the equation:
Fs = μs * (m * g)
Since Fs = Fc, we can equate the two equations:
μs * (m * g) = m * ac
Now, we can solve for the coefficient of static friction (μs):
μs = (m * ac) / (m * g)
Simplifying further:
μs = ac / g
Now, let's substitute the known values into the equation and perform the calculation.
To find the coefficient of static friction between the coin and the turntable, we can use the following steps:
1. Identify the relevant equations:
- Centripetal force equation: Fc = m * ac
- Frictional force equation: Fs = μs * N
2. Find the acceleration of the coin:
- The coin is rotating at a constant rate, so its acceleration is the centripetal acceleration.
- The centripetal acceleration can be calculated using the equation ac = v^2 / r, where v is the speed of the coin and r is the distance from the center of the turntable to the coin.
- Plugging in the given values, we have ac = (100 cm/s)^2 / 33 cm.
3. Calculate the centripetal force:
- The centripetal force is provided by the static friction between the coin and the turntable.
- The centripetal force can be calculated using the equation Fc = m * ac, where m is the mass of the coin.
- Rearranging the equation, we have Fc = m * (v^2 / r).
4. Determine the normal force:
- The normal force, N, is the force exerted by the turntable on the coin perpendicular to its surface.
- In this case, the normal force is equal to the weight of the coin.
- The weight, W, can be calculated using the equation W = m * g, where g is the acceleration due to gravity.
- Plugging in the given value g = 980 cm/s^2, we can find W.
5. Substitute the values into the frictional force equation:
- The frictional force is given by Fs = μs * N.
- By substituting Fs = Fc and N = W into the equation, we get μs * W = m * (v^2 / r).
6. Calculate the coefficient of static friction:
- Rearrange the equation to solve for the coefficient of static friction, μs.
- We have μs = (m * (v^2 / r)) / W.
7. Plug in the given values:
- Substitute the given values of m = mass of the coin, v = 100 cm/s, r = 33 cm, and W = weight of the coin into the equation.
- Calculate the coefficient of static friction.
By following these steps, you should be able to find the coefficient of static friction between the coin and the turntable.