Angular velocity (S)

A coin of mass m is placed a distance R from the center of a record player turntable. The coefficient of static friction is μs. The angular speed of the turntable is slowly increased to a value of ω0, at which point the coin slides off. Find ω0.
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Hint

Identify all the forces acting on the coin.
 

Solution

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Solution

Graphic - No title - coinplayer
Firstly, we ignore the tangential acceleration because we are told it is small.
  • Newton’s second law in the vertical direction gives

    \[ F_N - mg = 0 \hspace{0.5cm}\mbox{\rm so}\hspace{0.5cm} F_N = mg \]

  • Newton’s second law in the horizontal direction gives S2.10

    \[ F_F = \frac{mv^2}{r} = mr\omega_0 ^2 \]
    making use of Key Point 5.10.

    On the point of slipping, the frictional force is given by (S2.8)

    FF=μsFN
    Combining these results we find
    \[ \mu_s mg = mr\omega_0 ^2 \hspace{0.5cm}\mbox{\rm so}\hspace{0.5cm} \omega_0 = \sqrt{\frac{\mu_s g}{r}} \]