The tires of a car make 75 revolutions as the car reduces its speed uniformly from 95 km/h to 55 km/h. The tires have a diameter of 0.80 m.

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. We'll take care of a bunch of unit conversion stuff first and then we'll answer the questions. So the radius of this car tyre is 0.80 meters diameter divided by 2 which is 0.40 meters; the initial velocity of the car is 95 kilometers per hour which we convert into 26.39 meters per second by multiplying by 1 hour for every 3600 seconds then multiplying by 1000 meters for every kilometer and same for the final velocity, 55 kilometers per hour divide by 3.6 is essentially what's happening here and that gives 15.278 meters per second and then converting those into angular velocities: for the initial angular velocity is v i over r and that's 26.39 meters per second divided by radius of 0.40 meters 65.98 radians per second; and for the final angular velocity— same idea—and you get 38.194. The angular displacement of 75 revolutions expressed in radians by multiplying it by 2π radians for every revolution is 471.2 radians and now we turn our attention to the questions. What was the angular acceleration of the tyres? So the final angular velocity squared is the initial angular velocity squared plus 2 times the angular acceleration times the angular displacement. And we can subtract ω i squared from both sides and switch the sides around and get 2αΘ equals ω f squared minus ω i squared. And then divide both sides by 2 times Θ and you get that the angular acceleration is the final angular velocity squared minus the initial angular velocity squared divided by 2 times the angular displacement. So that's 38.194 radians per second squared minus 65.98 radians per second squared divided by 2 times 471.2 radians and we get negative 3.1 radians per second squared is the angular acceleration of the tyres. We can still say the word acceleration instead of deceleration if we like because we have this negative acceleration is the same as saying deceleration. So for part (b), we are asked how much more time is required for it to stop? So it says the word 'more' there so that means after its reached the 55 kilometer per hour, how much additional time is required to go from 55 kilometers an hour to 0 or in other words from 38.194 radians to 0? So we use this formula here and we are gonna solve for the time knowing that the final angular velocity is zero and we'll subtract ω i from both sides and then also divide both sides by α and we get t is negative ω i over α and I'm hoping that I don't confuse you too much with the subscripts because in this case, the initial angular velocity is 38.194 but I labeled it ω f here because it depends which time frame you are considering. So for part (a), the final moment is when it's going at 55 kilometers an hour or 38.194 but in part (b), this 38.194 is the moment when you begin considering and so that's why I have it replacing ω i. So this works out to 12 seconds after we divide that by the angular acceleration we figured out in part (a). And then the total distance traveled and that is the total distance traveled from the beginning velocity of 95 kilometers an hour and that means in this formula ω i is going to be 65.98 radians per second that we arrived at by you know, taking the 95 kilometers per hour, converting it into meters per second and dividing by the radius of 0.4. So we know that the final angular velocity is zero because the car comes to a stop and then subtract ω i squared from both sides and then divide both sides by 2Θ and you get Θ is negative ω i squared over 2Θ and this is gonna give us the angular displacement which does not answer the question. because we want to know the total distance traveled and so we have to convert this into a linear unit and so we'll multiply by the radius of the wheel. And so times by 0.80 meters divided by 2 times this 708.67 radians that gives about 280 meters traveled by the wheel in total from 95 kilometers per hour to 0.