A car starts from rest on a curve with a radius of 190m and accelerates at 0.600m/s^2 . How many revolutions will the car have gone through when the magnitude of its total acceleration is 3.50m/s^2 ?
You will need to consider both centripetal aceleration (a_r) and tangential acceleration (a_t), which are perpendicular.
3.50 = a_t^2 + a_c^2 = a_c^2 + 0.36
a_c^2 = V^2/R = 3.14 m/s^2
Solve for V when the acceleration of 3.50 m/s^2 is reached. From that, get the time and the number of revolutions
To find the number of revolutions the car will have gone through, we need to use the formula for centripetal acceleration and solve for the number of revolutions.
The formula for centripetal acceleration is:
a = (v² / r)
where:
a is the centripetal acceleration,
v is the velocity, and
r is the radius of the curve.
In this case, the car starts from rest, so the initial velocity (v₀) is 0 m/s.
The total acceleration (a) is given as 3.50 m/s².
The radius of the curve (r) is given as 190 m.
The car accelerates at a rate of 0.600 m/s².
To find the final velocity (v), we can use the equation:
v = v₀ + a₀t
where v₀ is the initial velocity, a₀ is the acceleration, and t is the time.
Since the car starts from rest, we can simplify the equation to:
v = a₀t
Now, we can substitute the values into the formula for centripetal acceleration and solve for t:
a = (v² / r)
3.50 = (v² / 190)
Now, rearranging the equation to solve for v:
v² = 3.50 * 190
v² = 665
Taking the square root of both sides:
v = √665
v ≈ 25.787 m/s
Now, we can use the equation v = a₀t to solve for t:
25.787 = 0.600t
Solving for t:
t ≈ 42.98 seconds
Finally, to find the number of revolutions, we need to calculate the angular displacement (θ) using the equation:
θ = (v₀t + (1/2)at²) / r
Since the car starts from rest, the initial angular displacement (θ₀) is 0.
Substituting the values:
θ = (0.5 * 0.600 * (42.98)²) / 190
θ ≈ 2.195 radians
Now, since there are 2π radians in one revolution, we can calculate the number of revolutions:
Number of revolutions = θ / (2π)
Number of revolutions = 2.195 / (2π)
Number of revolutions ≈ 0.349 revolutions
Therefore, the car will have gone through approximately 0.349 revolutions when the magnitude of its total acceleration is 3.50 m/s².