As you drive down the road at 21 m/s you press on the gas pedal and speed up with a uniform acceleration of 1.24 m/s2 for 0.60 s. If the tires on your car have a radius of 33 cm, what is their angular displacement during this period of acceleration? in radians
To find the angular displacement of the tires, we first need to determine the linear displacement of the car during the period of acceleration.
We can use the equation of motion:
\(v = u + at\)
Where:
\(v\) is the final velocity (unknown),
\(u\) is the initial velocity (21 m/s),
\(a\) is the acceleration (1.24 m/s²), and
\(t\) is the time interval (0.60 s).
Rearranging the equation to solve for \(v\), we have:
\(v = u + at\)
\(v = 21 + (1.24)(0.60)\)
Calculating the value of \(v\):
\(v = 21 + 0.744\)
\(v = 21.744\) m/s
Next, we can calculate the linear displacement using the formula:
\(s = ut + \frac{1}{2}at^2\)
Where:
\(s\) is the linear displacement (unknown),
\(u\) is the initial velocity (21 m/s),
\(a\) is the acceleration (1.24 m/s²), and
\(t\) is the time interval (0.60 s).
Plugging in the known values:
\(s = (21)(0.60) + \frac{1}{2}(1.24)(0.60)^2\)
Calculating the value of \(s\):
\(s = 12.6 + 0.4464\)
\(s = 13.0464\) meters
Now, we can calculate the angular displacement of the tires. The linear displacement is equal to the circumference of the circle traced by the tires during this period, which is given by:
\(2\pi r\)
Where:
\(r\) is the radius of the tires (33 cm = 0.33 m).
Plugging in the value of \(r\):
\(2\pi r = 2\pi(0.33)\)
Calculating the value of \(2\pi r\):
\(2\pi r = 2\pi (0.33)\)
\(2\pi r = 2.08\) meters
Since the linear displacement is equal to the arc length of the circular path, we can find the angular displacement by using the formula:
\(\theta = \frac{s}{r}\)
Where:
\(\theta\) is the angular displacement (unknown),
\(s\) is the linear displacement (13.0464 m), and
\(r\) is the radius of the tires (0.33 m).
Plugging in the known values:
\(\theta = \frac{13.0464}{0.33}\)
Calculating the value of \(\theta\):
\(\theta = 39.559\) radians
Therefore, the angular displacement of the tires during this period of acceleration is approximately 39.559 radians.