A 1.74 kg toy race car is slowing down while travelling around a circular horizontal track. If the track has a diameter of 0.690 m and the coefficient of rolling friction between the track and the race car is 0.180, what is the magnitude of angular acceleration?

To solve this problem, we need to use the following formulas:

1. The force of friction (Ff) = μN, where μ is the coefficient of friction and N is the normal force. In this case, because the car is moving in a horizontal circle, the normal force is equal to the weight of the car (mass * gravity). So, Ff = μmg.

2. The net force acting on the car is the centripetal force, which is equal to the force of friction. So, Fc = Ff. The formula for centripetal force is Fc = m * a_c, where m is the mass of the car and a_c is the centripetal acceleration.

3. The centripetal acceleration (a_c) is related to the angular acceleration (α) by the formula a_c = αr, where r is the radius of the circle.

Let's solve the problem step by step:

Step 1: Calculate the force of friction (Ff) Ff = μmg = 0.180 * 1.74 kg * 9.8 m/s² = 3.06 N

Step 2: Set the force of friction equal to the centripetal force to find the centripetal acceleration (a_c) Fc = Ff m * a_c = Ff a_c = Ff / m = 3.06 N / 1.74 kg = 1.76 m/s²

Step 3: Use the formula for centripetal acceleration to find the angular acceleration (α) a_c = αr α = a_c / r = 1.76 m/s² / (0.690 m / 2) = 5.11 rad/s²

So, the magnitude of the angular acceleration of the toy race car is 5.11 rad/s².