(a) Calculate the linear acceleration of a car, the 0.280 m radius tires of which have an angular acceleration of 15.0 rad/s2. Assume no slippage. (m/s2)
(b) How many revolutions do the tires make in 2.50 s if they start from rest? (rev)
(c) What is their final angular velocity? (rad/s)
(d) What is the final velocity of the car? (m/s)
(a) Well, you know what they say - no slippage, no problem! To calculate the linear acceleration of the car, we need to use the formula a = r * α, where a is the linear acceleration, r is the radius of the tires, and α is the angular acceleration. So, plugging in the values we have, a = 0.280 m * 15.0 rad/s^2 = 4.20 m/s^2.
(b) Revolutions? Oh, we're going in circles now? Alright, let's calculate how many revolutions the tires make in 2.50 seconds. Now, if they start from rest, we can use the formula θ = 1/2 * α * t^2, where θ is the angle covered, α is the angular acceleration, and t is the time. Since the tires start from rest, the initial angular velocity is 0. So, θ = 1/2 * 15.0 rad/s^2 * (2.50 s)^2 = 46.875 rad.
But wait, we want revolutions, not radians! One revolution is equal to 2π radians, so we can convert the angle to revolutions by dividing it by 2π. So, the tires make 46.875 rad / (2π rad/rev) ≈ 7.480 revolutions.
(c) To find the final angular velocity of the tires, we can use the formula ω = ω₀ + α * t, where ω is the final angular velocity, ω₀ is the initial angular velocity (which is 0 since they start from rest), α is the angular acceleration, and t is the time. Plugging in the values, we get ω = 0 + 15.0 rad/s^2 * 2.50 s = 37.5 rad/s.
(d) Finally, let's find the final velocity of the car. Using the formula v = ω * r, where v is the final velocity, ω is the final angular velocity, and r is the radius of the tires, we can calculate it. Plugging in the values, v = 37.5 rad/s * 0.280 m = 10.5 m/s.
So, the linear acceleration of the car is 4.20 m/s^2, the tires make approximately 7.480 revolutions, their final angular velocity is 37.5 rad/s, and the final velocity of the car is 10.5 m/s.
(a) To calculate the linear acceleration of the car, we can use the relationship between linear and angular acceleration. The linear acceleration (a) is related to the angular acceleration (α) and the radius of the tires (r) by the equation:
a = α * r
Given that the angular acceleration (α) is 15.0 rad/s^2 and the radius of the tires (r) is 0.280 m, we can substitute these values into the equation to find the linear acceleration.
a = 15.0 rad/s^2 * 0.280 m = 4.2 m/s^2
Therefore, the linear acceleration of the car is 4.2 m/s^2.
(b) If the tires start from rest, we can find the number of revolutions they make in a given time period using the relationship between angular acceleration (α) and the number of revolutions (rev) given by the equation:
rev = (1/2) * α * t^2 / π
In this case, the angular acceleration (α) is 15.0 rad/s^2 and the time period (t) is 2.50 s. Substituting these values into the equation, we can calculate the number of revolutions.
rev = (1/2) * 15.0 rad/s^2 * (2.50 s)^2 / π ≈ 18.85 revolutions
Therefore, the tires make approximately 18.85 revolutions in 2.50 s.
(c) The final angular velocity (ω) can be calculated using the relationship between angular acceleration (α) and angular velocity (ω) by the equation:
ω = α * t
Given that the angular acceleration (α) is 15.0 rad/s^2 and the time period (t) is 2.50 s, we can substitute these values into the equation to find the final angular velocity.
ω = 15.0 rad/s^2 * 2.50 s = 37.5 rad/s
Therefore, the final angular velocity of the tires is 37.5 rad/s.
(d) To find the final velocity of the car, we can use the relationship between linear velocity (v) and angular velocity (ω) given by the equation:
v = ω * r
Given that the final angular velocity (ω) is 37.5 rad/s and the radius of the tires (r) is 0.280 m, we can substitute these values into the equation to find the final velocity.
v = 37.5 rad/s * 0.280 m ≈ 10.5 m/s
Therefore, the final velocity of the car is approximately 10.5 m/s.
Correction:
b. d=0.5a*t^2 = 0.5*4.20*2.5^2=13.125 m
Revs = 13.125m * 1rev/1.76m = 7.46
c. V = 7.46rev/2.5s * 6.28rad/rev=18.73
rad/s.
d. V=18.73rad/s * 1.76m/6.28rad=5.25m/s.
a. Circumference=pi*2r = 3.14 0.560=1.76
m.
a=15rad/s^2 * 1.76m/6.28rad = 4.20 m/s^2
b. V = a*t = 4.20m/s^2 * 2.50s=10.51 m/s
d = 0.5a*t^2 = 0.5*10.51*2.5^2 = 32.84 m
Revs = 32.84m * 1rev/1.76m = 18.7
c. V = 18.7rev/2.5s * 6.28rad/rev = 46.9
rad/s.
d. V = 46.9rad/s * 1.76m/6.28rad = 13.14
m/s.