Question

The speedometer readings in passenger cars are based on angular speed of the car's tire axle. Because car's tires are filled with air [typically to the pressure of about 30 psi (pounds-square-inch)], the actual tire radius can be a bit different depending on the air temperature. You want to understand the impact of the air temperature on the tire radius and thus on the accuracy of the speedometer reading based on the ideal gas law. To simplify the calculation, you ignore the elasticity of the tire itself. By this assumption, if there is less air in the tire or the temperature is lower, the tire radius will become smaller keeping the air pressure in the tire unchanged (in other words, the air pressure is set entirely by the weight of the car). The tires of your car are most commonly used passenger car tires with specification of "P 215 / 65 R 15". The meaning of the specification (at the recommended tire pressure) is shown in the figure below. kW P215/65 R 15 rim diameter, 2R, = 15” = 0.381 m construction type. Radial type sidewall aspect ratio (65%); RH-R = 0.65 W=0.140 m tire width, W=0.215 m → tire type, Passenger vehicle Ry at temperature Ty at temperature Tc <TH For a further simplicity, you assume that the tire is always perfectly round and that the width of the tire is always constant at W = 0.215 m, as illustrated in the figure above. On one summer day you bring your car to a shop for a speedometer tuning. The speedometer is accurately tuned with a tire radius RH and the air temperature is The (a) The tires are perfectly sealed and the amount of the air in the tires does not change, and the season changes. On one chilly day you want to find out what the tire radius R was on the summer day when the speedometer was tuned. The air temperature on a cold day is Tc = 2.80 °C and the tire radius is Rc = 0.3165 m, if the actual speed of your car is 66.4 mph (miles-per-hour) when the speedometer reading is 68.8 mph, what must be the value of RH? RH = (b) What must be the air temperature TH on the summer day when the speedometer was tuned? TH = °C

Answer 1

Hey!

This is simple question you just have to read the question carefully 2-3 times.

given that at temp T(c) = 2.80 degree actual speed is 66.4 mph but the speedometer speed is 68.8 mph

and speedometer reading is based on the rotation of axle. and given that speedometer is tuned in summer at temp T(h). that means the distance travel in summer in a particular time by the tyre is calculated and according to that speedometer is tuned.

when the speedometer reading is 68.8 mph that means if temp was T(h) then this 68.8 reading is true reading. radius at T(h) is R(h). let at this speed rpm of axle = N rpm

so speed = 2*pi*R(h)*N/60 = 68.8 mph...................................................equation 1

but the actual speed is 66.4 because of temp is T(c) which is lower than the T(h) so now the radius is R(c).

so speed is = 2*pi*R(c)*N/60 = 66.4 mph..............................................equation 2

divide equation 1 by 2

R(h)/R(c) = 68.8/66.4

R(h) = 0.3279 m

now given that tyre is sealed and due to temp change only radius is changed (pressure, width are constant)

according to ideal gass equation

P*V = mRT

is tyre is sealed mass of air will remain constant

P*V(h)/R*T(h) = P*V(c)/R*T(c)

V(h)/T(h) = V(c)/T(c) ........................................................equation 3

V(h) = pi*[R(h) - R(rim)]^2*(width)

V(c) = pi*[R(c) - R(rim)]^2*(width)

putting values in equation 3

[R(h) - R(rim)]^2 / [R(c) - R(rim)]^2 = T(h)/T(c)

R(rim) =0.1905 m

putting all the values

1.1891 = T(h)/T(c)

T(h) = 1.1891*2.80 = 3.3295 degree