Hi there!
(A)
For a banked curve, we know that:
∑F = MgsinФ (force due to gravity)
And:
∑F = mv²/r
The cosine of the centripetal force, mv²/r, is involved in the summation of forces, so:
(mv²/r)cosФ = mgsinФ
Simplify the equation:
(v²/r)cosФ = gsinФ
Divide both sides by cosФ:
(v²/r) = gtanФ
Solve for v:
v = √grtanФ
Plug in given values. g ≈ 9.8 m/s²
v = √(9.8 · 120 · tan(15)) = 17.75 m/s
(b)
Incorporating friction, we now can rewrite the sum of forces. Let's first determine the frictional force acting on the car.
As the horizontal component of the centripetal force is the net force, the vertical component works in the direction of the normal force, so we must include it in the calculation of the frictional force.
First, convert 15 km/h to m/s:
15km / h * 1000 m / 1km * 1 h / 3600 sec = 4.167 m/s
Thus:
∑F = mgsinФ - μ(mgcosФ + (mv²/r)sinФ)
mv²/r = mgsinФ - μ(mgcosФ + (mv²/r)sinФ)
Divide all terms by the mass:
v²/r = gsinФ - μ(gcosФ + (v²/r)sinФ)
Solve for μ:
v²/r - gsinФ = - μ(gcosФ + (v²/r)sinФ)
-(v²/r - gsinФ)/(gcosФ + (v²/r)sinФ) = μ
Plug in the given values:
μ = 0.25