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1 Friction and Power

The apparatus is the Prony brake: a wall-mounted wheel with a rope that runs along a groove in the wheel perimeter. The rope tension can be adjusted with two string scales. Additional equipment includes a stop-watch and a tape-measure.

When the Prony brake wheel is rotated at constant angular velocity $ω$ , the force of friction between wheel and rope causes the rope tensions on the two sides of the wheel to become un-equal. The situation is illustrated below, with a small segment of rope that subtends an angle $2 d θ$ on the wheel. When the wheel is rotated counter-clockwise, the tension at $θ + 2 d θ$ exceeds that at $θ$ ; the force-balance equations are

d N = T (θ) sin d θ + T (θ + 2 d θ) sin d θ, T (θ) cos d θ + d F_{f} = T (θ + 2 d θ) cos d θ

The normal force and frictional force on this little segment are differential quantities; both vanish as $d θ \to 0$ .
Using the small angle approximation these become

d N = (T (θ) + T (θ + 2 d θ)) d θ \approx 2 T (θ) d θ, d F_{f} = T (θ + 2 d θ) - T (θ) \approx 2 \frac{d T (θ)}{d θ} d θ

Dividing these two equations we ﬁnd that

\frac{d F_{f}}{d N} = μ_{k} = \frac{1}{T (θ)} \frac{d T (θ)}{d θ}

Notice that the two ropes lose contact with the wheel at $θ = 0, π$ , so integrate;

μ_{k} \int_{0}^{π} d θ = μ_{k} π = \int_{0}^{π} \frac{1}{T (θ)} \frac{d T (θ)}{d θ} d θ = \frac{1}{2} \int_{T (0)}^{T (π)} \frac{d T}{T} = ln \frac{T (π)}{T (0)} = ln \frac{T_{1}}{T_{2}}

We arrive at a formula for the coefficient of kinetic friction, valid for any rotational speed $ω$

μ_{k} = \frac{1}{π} ln \frac{T_{1}}{T_{2}}, T_{1} > T_{2}

This provides us with a simple means of accurately measuring the coefficient of friction.

In addition the Prony brake can be used to measure the power output of whoever turns the wheel.
The frictional force on a segment of rope (subtending $d θ$ ) in contact with the wheel is

d F_{f} (θ) = d T (θ)

As the wheel turns through arc length $s$ , the work done by friction in this segment is

d W = s d F_{f} (θ) = s d T (θ)

Add up the work done by the forces of friction on each segment of the rope in contact with the wheel

W = s \int d T (θ) = \int_{0}^{π} \frac{d T (θ)}{d θ} d θ = s (T (π) - T (0)) = s (T_{1} - T_{2})

Differentiate with respect to time to get the power expended to turn the wheel at $ω$ , maintaining tension-difference $T_{1} - T_{2}$ ;

P = \frac{d s}{d t} = (T_{1} - T_{2}) = R ω (T_{1} - T_{2})

1.1 Experimental procedure

Determine the coefficient of friction $μ_{s}$ between the rope and wheel of the Prony brake. The formula derived for $μ_{k}$ is valid for $μ_{s}$ ; see how much of a tension-difference you can create before static friction is overcome, and the rope slips.
Determine the kinetic coefficient by turning the wheel at a constant rate, and again record $T_{1}$ and $T_{2}$ .

Determine the maximal power that your arms can deliver by rotating the wheel as fast as you can for $60 s$ , while someone counts revolutions. Convert this into horsepower. Are you disappointed with the results?

Determine the maximal power that your legs can deliver by measuring a ﬂight of stairs in the corridor and have your lab partner time you as you climb them as fast as you can. This should be less disappointing. Be glad that you don’t have to walk on your hands all day long.

1.2 Pre-lab questions

1. What force prevents a knot from slipping? Explain yourself.

2. You rotate the Prony brake wheel through $90$ revolutions in $60 s$ . The Deluxe Model has $R = 0.6 m$ , and during this time $T_{1} = 88 N$ and $T_{2} = 28 N$ . What is your power output?

2. After a hearty breakfast a physics student weighs in at $62 k g$ . This kid climbs the stairs of Van Vleck Hall in Madison, the Math building, all $13$ ﬂoors, in $24 s$ . What is her power output? Each ﬂoor is $3.5 m$ high.
Which scares you more, the awesome power or the fact that there are thirteen ﬂoors in the Math building alone?

1.3 Lab report

Power and friction


Experimenter 1		Experimenter 2

Experimenter 3		Experimenter 4


Prony brake setup

Radius $R =$	Resting $T_{1} =$	Resting $T_{2} =$

Coefficients of friction




Static	$T_{1} =$	$T_{2} =$	$μ_{s} =$



Kinetic	$T_{1} =$	$T_{2} =$	$μ_{k} =$

Power




Leg-power	height $H =$	Time $T =$	Power $P =$



Arm-power	Turns $N =$	Time $T =$	Power $P =$