Question

An automobile engine can produce 200 N· m of torque. Calculate the angular acceleration produced if 95.0% of this torque is applied to the drive shaft, axle, and rear wheels of a car, given the following information. The car is suspended so that the wheels can turn freely. Each wheel acts like a 15.0-kg disk that has a 0.180-m radius. The walls of each tire act like a 2.00-kg annular ring that has inside radius of 0.180 m and outside radius of 0.320 m. The tread of each tire acts like a 10.0-kg hoop of radius 0.330 m. The 14.0-kg axle acts like a rod that has a 2.00-cm radius. The 30.0-kg drive shaft acts like a rod that has a 3.20-cm radius.

(a) 2.16 rad/s²
(b) 2.40 rad/s²
(c) 2.55 rad/s²
(d) 2.80 rad/s²

Answer 1

To find the angular acceleration in a physics problem involving a car engine and various components like wheels and axles, calculate the total moment of inertia and then divide the effective torque by this total inertia.

Step-by-step explanation:

To calculate the angular acceleration produced by the torque in an automobile engine, we use the rotational dynamics equation α = τ / I, where α is the angular acceleration, τ is the torque, and I is the moment of inertia. The torque applied to the drive shaft, axle, and rear wheels of the car is 95% of 200 N·m, which equals 190 N·m. The moment of inertia for each component is calculated using standard formulas for rigid bodies:

• Each wheel (disk): I = (1/2)mr²
• Each tire's wall (annular ring): I = m*(r_o² + r_i²)//2
• Tread of each tire (hoop): I = mr²
• Axle (rod about center): I = (1/2)mr²
• Drive shaft (rod about center): I = (1/2)mr²

After calculating the individual moments of inertia for each component, the total moment of inertia I_total is the sum of all the components' moments of inertia. Finally, the angular acceleration is calculated by dividing the total torque by the total moment of inertia. This physics problem requires knowledge of rotational dynamics, moment of inertia, and the use of calculus to find the sum of moments for different shapes. The exact numerical answer would follow these steps but the calculation is not shown here.