Question

A car at the Indianapolis-500 accelerates uniformly from the pit area, going from rest to 340 km/h in a semicircular arc with a radius of 196 m.

Determine the tangential acceleration of the car when it is halfway through the turn, assuming constant tangential acceleration.

Determine the radial acceleration of the car at this time.

If the curve were flat, what would the coefficient of static friction have to be between the tires and the roadbed to provide this acceleration with no slipping or skidding?

Answer 1

[AI-generated answer — Explain Bot]

To determine the tangential acceleration of the car when it is halfway through the turn, we can use the formula for tangential acceleration:

𝑎𝑡 = 𝑟 × 𝑎𝑛𝑔𝑙𝑒

where 𝑎𝑡 is the tangential acceleration, 𝑟 is the radius of the arc, and 𝑎𝑛𝑔𝑙𝑒 is the angular acceleration of the car.

We need to find the angular acceleration of the car first. To do this, we need to know the change in angular velocity of the car.

The car starts from rest and accelerates to a speed of 340 km/h, which is 340,000 m/3600 s = 94.44 m/s. Since this happens over a semicircular arc, the angular displacement is π radians (half a revolution).

The formula for angular acceleration is:

𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = (𝑣𝑓^2 - 𝑣𝑖^2) / (2 × 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡)

where 𝑣𝑓 is the final angular velocity, 𝑣𝑖 is the initial angular velocity, and 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 is the angular displacement.

Since the car starts from rest, the initial angular velocity 𝑣𝑖 is 0.

Using the values provided, we can calculate the angular acceleration:

𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = (𝑣𝑓^2 - 𝑣𝑖^2) / (2 × 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡)
= (94.44^2 - 0^2) / (2 × π)
= 28212.96 / (2 × π)
≈ 4489.47 rad/s²

Now that we have the angular acceleration, we can calculate the tangential acceleration using the formula mentioned earlier:

𝑎𝑡 = 𝑟 × 𝑎𝑛𝑔𝑙𝑒
= 196 m × 4489.47 rad/s²
≈ 879,304.12 m/s²

Therefore, the tangential acceleration of the car when it is halfway through the turn is approximately 879,304.12 m/s².

To determine the radial acceleration of the car at this time, we can use the formula:

𝑎𝑟 = 𝑣^2 / 𝑟

where 𝑎𝑟 is the radial acceleration, 𝑣 is the velocity of the car, and 𝑟 is the radius of the arc.

At halfway through the turn, the speed of the car would be halfway between 0 and 340 km/h. So the velocity would be 170 km/h, which is 170,000 m/3600 s = 47.22 m/s.

Using the given radius of 196 m, we can calculate the radial acceleration:

𝑎𝑟 = 𝑣^2 / 𝑟
= 47.22^2 / 196
= 1126.80 m/s²

Therefore, the radial acceleration of the car at halfway through the turn is approximately 1126.80 m/s².

Finally, to find the coefficient of static friction needed between the tires and the roadbed, we need to consider the maximum possible static friction force between the tires and the roadbed. This is given by:

𝑓𝑠 = 𝑚𝑢𝑠𝑙𝑖𝑑𝑒𝑠 𝑓𝑟

where 𝑓𝑠 is the maximum static friction force, 𝑚 is the mass of the car, 𝑢𝑠 is the coefficient of static friction, and 𝑓𝑟 is the radial force on the car.

The radial force on the car can be calculated using Newton's second law of motion:

𝑓𝑟 = 𝑚 × 𝑎𝑟

Given that the mass 𝑚 of the car is not provided, let's assume it to be 1000 kg.

Therefore,

𝑓𝑟 = 1000 kg × 1126.80 m/s²
= 1,126,800 N

To avoid slipping or skidding, the maximum static friction force 𝑓𝑠 should be equal to 𝑓𝑟:

𝑓𝑠 = 𝑓𝑟
𝑚𝑢𝑠𝑙𝑖𝑑𝑒𝑠 𝑓𝑟 = 𝑓𝑟
𝑚𝑢𝑠𝑙𝑖𝑑𝑒𝑠 × 1,126,800 N = 1,126,800 N

Canceling out the force term, we can solve for the coefficient of static friction 𝑢𝑠:

𝑢𝑠𝑙𝑖𝑑𝑒𝑠 = 1

So, to provide the given acceleration with no slipping or skidding, the coefficient of static friction between the tires and the roadbed would need to be equal to 1.