Question

Problem 4. Two carts are released simultaneously from rest at the top of a ramp of height 3.0 m, and they roll without slipping to a finish line at the bottom. Each cart has four wheels which are uniform disks of mass .50 kg. Cart A has total mass 10.0 kg (including the wheels) and each of its wheels have radius 4.0 cm. Cart B has the same total mass as cart A but its wheels have double the mass and double the radius. (a) Which cart wins the race? Answer: Cart A (b) How fast is the winning car moving when it reaches the finish line? Answer should be 7.31 m/s

Answer 1

inital energy at top Ei = M*g*h = 10*9.8*3 = 294 J

final energy of at the bottom = (1/2)*I*w^2 + (1/2)*M'*v^2

moment of inertia of wheels I = 4*[ (1/2)*m*R^2 + mR^2 ] = 6*m*R^2

m = mass of wheel

R = radis of wheel

w =angular speed

v = linear speed

M' = mass of the cart

w = v/R

M' = M - 4m

Ef = 3*m*v^2 + (1/2)*M*v^2 - 2*m*V^2

Ef = (m + M/2)*v^2

from energy conservation

Ef = Ei

(m + M/2)*v^2 = M*g*h

v = sqrt( Mgh/(m+M/2) )

mass of wheel of A is samller than wheels of B

speed of A more than B

A wins the race

======================

speed of A = sqrt((10*9.8*3)/(0.5 + 5)) = 7.31 m/s