The tires of a car make 85 revolutions as the car reduces its speed uniformly from 110 km/h to 40 km/h. The tires have a diameter of 0.80 m. (a) What was the angular acceleration?

To solve this problem, we need to follow these steps:

Step 1: Convert the speeds from km/h to m/s The initial speed (u) = 110 km/h = 110 * 1000/3600 = 30.56 m/s The final speed (v) = 40 km/h = 40 * 1000/3600 = 11.11 m/s

Step 2: Calculate the total distance travelled The total distance travelled is the number of revolutions times the circumference of the tires. The circumference (C) of a tire is given by C = πd, where d is the diameter of the tire. So, C = π * 0.80 = 2.51 m The total distance (s) = number of revolutions * circumference = 85 * 2.51 = 213.35 m

Step 3: Use the equation of motion to find the time taken We can use the equation v = u + at to find the time taken (t) for the car to reduce its speed. But first, we need to find the acceleration (a). We can rearrange the equation to find a = (v - u) / t. But we don't have the time (t). So, we can use another equation of motion that doesn't require time: v^2 = u^2 + 2as. We can rearrange this to find a = (v^2 - u^2) / (2s). So, a = (11.11^2 - 30.56^2) / (2 * 213.35) = -1.35 m/s^2

Step 4: Find the angular acceleration The angular acceleration (α) is the linear acceleration (a) divided by the radius (r) of the tire. The radius is half the diameter, so r = 0.80 / 2 = 0.40 m. So, α = a / r = -1.35 / 0.40 = -3.375 rad/s^2

So, the angular acceleration of the car as it reduced its speed was -3.375 rad/s^2. The negative sign indicates that it was a deceleration, i.e., the car was slowing down.

To solve this problem, we need to follow these steps:

Step 1: Convert the speeds from km/h to m/s

• Initial speed (u) = 110 km/h = 110 * 1000/3600 = 30.56 m/s
• Final speed (v) = 40 km/h = 40 * 1000/3600 = 11.11 m/s

Step 2: Calculate the angular speed

• The angular speed (ω) is the linear speed (v) divided by the radius (r) of the tire. The radius is half the diameter, so r = 0.80/2 = 0.40 m.
• Initial angular speed (ω1) = u/r = 30.56/0.40 = 76.4 rad/s
• Final angular speed (ω2) = v/r = 11.11/0

To solve this problem, we need to use the equations of motion.

Step 1: Convert the speeds from km/h to m/s Initial speed (u) = 110 km/h = 110 * 1000/3600 = 30.56 m/s Final speed (v) = 40 km/h = 40 * 1000/3600 = 11.11 m/s

Step 2: Calculate the total distance travelled The distance travelled by the car is equal to the circumference of the tire times the number of revolutions. The circumference (C) of a circle is given by C = πd, where d is the diameter of the circle. So, the distance (s) is: s = number of revolutions * circumference = 85 * π * 0.80 = 213.6283 m

Step 3: Use the equation of motion to find the time We can use the equation v = u + at to find the time (