A small car with mass m and speed 2v and a large car with mass 2m and speed v both travel the same circular section of an unbanked road. if the frictional force required to keep the small car on the road without skidding is f, then the frictional force required to keep the large car on the road without skidding is?

Question: A small car with mass m and speed 2v and a large car with mass 2m and speed v both travel the same circular section of an unbanked road. if the frictional force required to keep the small car on the road without skidding is f, then the frictional force required to keep the large car on the road without skidding is?

The frictional force required to keep a car traveling in a circular path without skidding is given by:

f = m*v^2/R

where m is the mass of the car, v is its speed, and R is the radius of the circular path.

For the small car with mass m and speed 2v, the frictional force required is:

f = m*(2v)^2/R = 4mv^2/R

For the large car with mass 2m and speed v, the frictional force required is:

f' = 2m*v^2/R

We can use the principle of conservation of energy to relate the speeds and radii of the circular paths for the two cars. The kinetic energy of each car is given by:

KE = (1/2)*m*v^2 for the small car

KE' = (1/2)*(2m)*v^2 = 2*(1/2)*m*v^2 = m*v^2 for the large car

Since the total energy is conserved, the sum of the kinetic and potential energies is constant for both cars. For a circular path of radius R, the potential energy is given by:

PE = m*g*R

where g is the acceleration due to gravity.

Therefore, the total energy of each car is:

E = KE + PE = (1/2)*m*v^2 + m*g*R for the small car

E' = KE' + PE = m*v^2 + m*g*R for the large car

Setting the total energies of the two cars equal to each other, we get:

(1/2)*m*v^2 + m*g*R = m*v^2 + m*g*R

Simplifying, we get:

(1/2)*m*v^2 = (1/2)*m*v^2

Therefore, the speeds of the two cars are the same, and we can equate the expressions for f and f' to get:

4mv^2/R = 2m*v^2/R

Simplifying, we get:

f' = 2f

Therefore, the frictional force required to keep the large car on the road without skidding is twice that required for the small car, given the same speed and radius of the circular path.

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