The blade of a windshield wiper moves through an angle of 90.0° in 0.31 s. The tip of the blade moves on the arc of a circle that has a radius of 0.59 m. What is the magnitude of the centripetal acceleration of the tip of the blade?
φ=90°=π/2
ω= φ/t
a= ω²R= φ²R/t²=π²•0.59/0.31²=60.6 m/s²
To find the magnitude of the centripetal acceleration of the tip of the blade, we can use the formula:
centripetal acceleration = (angular velocity)^2 * radius
First, we need to find the angular velocity. The angle covered by the blade is given as 90.0°. To convert this to radians, we multiply by π/180:
angle in radians = 90.0° * (π/180) = π/2 radians
Next, we need to calculate the angular velocity, which is the angle covered divided by the time taken:
angular velocity = (π/2 radians) / (0.31 s) = π/2/0.31 radians/s
Now, substituting the given radius of 0.59 m and the angular velocity, we can calculate the centripetal acceleration:
centripetal acceleration = ((π/2)/0.31 radians/s)^2 * 0.59 m
Calculating this expression, we get:
centripetal acceleration ≈ 1.86 m/s²
Therefore, the magnitude of the centripetal acceleration of the tip of the blade is approximately 1.86 m/s².
To find the magnitude of the centripetal acceleration, we can use the formula:
ac = (v^2) / r
Where ac is the centripetal acceleration, v is the linear velocity, and r is the radius of the circle.
To begin, we need to find the linear velocity. The angular displacement of the blade is given as 90.0°, and the time taken to cover this angular displacement is given as 0.31 s. Angular displacement is related to linear displacement by the formula:
s = rθ
Where s is the linear displacement, r is the radius, and θ is the angular displacement.
Substituting the given values, we get:
s = (0.59 m) × (90.0°) = 53.1 m
Next, we can find the linear velocity by dividing the linear displacement by the time taken:
v = s / t = 53.1 m / 0.31 s ≈ 171.0 m/s
Finally, we can plug this value of linear velocity and the given radius into the formula for centripetal acceleration:
ac = (v^2) / r = (171.0 m/s)^2 / 0.59 m ≈ 49,000 m/s^2
Therefore, the magnitude of the centripetal acceleration of the tip of the blade is approximately 49,000 m/s^2.