Riders in a carnival ride stand with their backs against the wall of a circular room of diameter \(8.0 \, \text{m}\). The room is spinning horizontally about an axis through its center at a rate of \(45 \, \text{rev/min}\) when the floor drops so that it no longer provides any support for the riders. What is the minimum coefficient of static friction between the wall and the rider required so that the rider does not slide down the wall? • Nerd Notes

First, convert the rotational speed to radians per second: [katex]\omega = 45 , \frac{\text{rev}}{\text{min}} \times \frac{1 , \text{min}}{60 , \text{s}} \times \frac{2\pi , \text{radians}}{1 , \text{rev}}[/katex].

Note you can convert to linear velocity [katex] v [/katex] instead of using [katex] \omega [/katex], since [katex] v = \omega r [/katex], where [katex] r [/katex] is the radius of the circle.

The minimum coefficient of static friction is found using the balance of forces in the vertical direction, with the static frictional force providing the upward force to counteract gravity, and the centripetal force being provided by the normal force.

Step	Formula / Derivation	Reasoning
1	[katex]a_c = \frac{v^2}{R}[/katex]	Centripetal acceleration formula.
2	[katex]v = R\omega[/katex]	Linear velocity in terms of radius and angular velocity.
3	[katex]a_c = R\omega^2[/katex]	Substitute [katex]v[/katex] into the centripetal acceleration formula.
4	[katex]F_c = m a_c = m R\omega^2[/katex]	Centripetal force provided by the normal force.
5	[katex]f_s = \mu_s N[/katex]	Static frictional force opposing gravity.
6	[katex]f_s = m g[/katex]	Static frictional force equals gravitational force to prevent sliding.
7	[katex]\mu_s = \frac{f_s}{N} = \frac{mg}{mR\omega^2}[/katex]	Substitute [katex]f_s[/katex] and [katex]N[/katex] with their equivalents.
8	[katex]\mu_s = \frac{g}{R\omega^2}[/katex]	Cancel [katex]m[/katex] from the equation.
9	[katex]\mu_s = \frac{g}{R\left(\frac{45}{60} \times 2\pi\right)^2}[/katex]	Substitute [katex]\omega[/katex] with its value in terms of rev/min to rad/s.
10	[katex]\mu_s = \frac{9.8}{4\left(\frac{3}{4} \pi\right)^2}[/katex]	Substitute given values for [katex]g[/katex] and [katex]R[/katex].
11	[katex]\mu_s = \frac{9.8}{4\left(\frac{9}{16} \pi^2\right)}[/katex]	Simplify the expression for [katex]\omega^2[/katex].
12	[katex]\boxed{\mu_s = \frac{9.8 \times 16}{4 \times 9 \pi^2}}[/katex]	Final expression for [katex]\mu_s[/katex].

Now let’s calculate the exact value for the minimum coefficient of static friction required.

	Linear speed	Centripetal acceleration	Tangential acceleration
\( (a) \)	\( R\omega \)	\( R\omega^{2} \)	\( R\alpha \)
\( (b) \)	\( R\omega \)	\( R\alpha \)	\( R\omega^{2} \)
\( (c) \)	\( R\omega^{2} \)	\( R\alpha \)	\( R\omega \)
\( (d) \)	\( R\omega \)	\( R\omega^{2} \)	\( R\omega \)
\( (e) \)	\( R\omega^{2} \)	\( R\alpha \)	\( R\omega^{2} \)

Linear speed

Centripetal acceleration

Tangential acceleration

\( (a) \)

\( R\omega \)

\( R\omega^{2} \)

\( R\alpha \)

\( (b) \)

\( R\omega \)

\( R\alpha \)

\( R\omega^{2} \)

\( (c) \)

\( R\omega^{2} \)

\( R\alpha \)

\( R\omega \)

\( (d) \)

\( R\omega \)

\( R\omega^{2} \)

\( R\omega \)

\( (e) \)

\( R\omega^{2} \)

\( R\alpha \)

\( R\omega^{2} \)

Kinematics	Forces
\(\Delta x = v_i t + \frac{1}{2} at^2\)	\(F = ma\)
\(v = v_i + at\)	\(F_g = \frac{G m_1 m_2}{r^2}\)
\(v^2 = v_i^2 + 2a \Delta x\)	\(f = \mu N\)
\(\Delta x = \frac{v_i + v}{2} t\)	\(F_s =-kx\)
\(v^2 = v_f^2 \,-\, 2a \Delta x\)

Circular Motion	Energy
\(F_c = \frac{mv^2}{r}\)	\(KE = \frac{1}{2} mv^2\)
\(a_c = \frac{v^2}{r}\)	\(PE = mgh\)
\(T = 2\pi \sqrt{\frac{r}{g}}\)	\(KE_i + PE_i = KE_f + PE_f\)
	\(W = Fd \cos\theta\)

Momentum	Torque and Rotations
\(p = mv\)	\(\tau = r \cdot F \cdot \sin(\theta)\)
\(J = \Delta p\)	\(I = \sum mr^2\)
\(p_i = p_f\)	\(L = I \cdot \omega\)

Simple Harmonic Motion	Fluids
\(F = -kx\)	\(P = \frac{F}{A}\)
\(T = 2\pi \sqrt{\frac{l}{g}}\)	\(P_{\text{total}} = P_{\text{atm}} + \rho gh\)
\(T = 2\pi \sqrt{\frac{m}{k}}\)	\(Q = Av\)
\(x(t) = A \cos(\omega t + \phi)\)	\(F_b = \rho V g\)
\(a = -\omega^2 x\)	\(A_1v_1 = A_2v_2\)

Constant	Description
[katex]g[/katex]	Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface
[katex]G[/katex]	Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex]
[katex]\mu_k[/katex] and [katex]\mu_s[/katex]	Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion.
[katex]k[/katex]	Spring constant, in [katex]\text{N/m}[/katex]
[katex] M_E = 5.972 \times 10^{24} , \text{kg} [/katex]	Mass of the Earth
[katex] M_M = 7.348 \times 10^{22} , \text{kg} [/katex]	Mass of the Moon
[katex] M_M = 1.989 \times 10^{30} , \text{kg} [/katex]	Mass of the Sun

Variable	SI Unit
[katex]s[/katex] (Displacement)	[katex]\text{meters (m)}[/katex]
[katex]v[/katex] (Velocity)	[katex]\text{meters per second (m/s)}[/katex]
[katex]a[/katex] (Acceleration)	[katex]\text{meters per second squared (m/s}^2\text{)}[/katex]
[katex]t[/katex] (Time)	[katex]\text{seconds (s)}[/katex]
[katex]m[/katex] (Mass)	[katex]\text{kilograms (kg)}[/katex]

Variable	Derived SI Unit
[katex]F[/katex] (Force)	[katex]\text{newtons (N)}[/katex]
[katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy)	[katex]\text{joules (J)}[/katex]
[katex]P[/katex] (Power)	[katex]\text{watts (W)}[/katex]
[katex]p[/katex] (Momentum)	[katex]\text{kilogram meters per second (kgm/s)}[/katex]
[katex]\omega[/katex] (Angular Velocity)	[katex]\text{radians per second (rad/s)}[/katex]
[katex]\tau[/katex] (Torque)	[katex]\text{newton meters (Nm)}[/katex]
[katex]I[/katex] (Moment of Inertia)	[katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex]
[katex]f[/katex] (Frequency)	[katex]\text{hertz (Hz)}[/katex]

Prefix	Symbol	Power of Ten
Pico-	p	[katex]10^{-12}[/katex]
Nano-	n	[katex]10^{-9}[/katex]
Micro-	µ	[katex]10^{-6}[/katex]
Milli-	m	[katex]10^{-3}[/katex]
Centi-	c	[katex]10^{-2}[/katex]
Deci-	d	[katex]10^{-1}[/katex]
(Base unit)	–	[katex]10^{0}[/katex]
Deca- or Deka-	da	[katex]10^{1}[/katex]
Hecto-	h	[katex]10^{2}[/katex]
Kilo-	k	[katex]10^{3}[/katex]
Mega-	M	[katex]10^{6}[/katex]
Giga-	G	[katex]10^{9}[/katex]
Tera-	T	[katex]10^{12}[/katex]

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