A car starts from rest on a circular curve with a radius of 120 m and accelerates tangentially at 1.0 m/s2
.
Through what angle will the car have travelled when the magnitude of its total acceleration is 2.0 m/s2
total a = v^2/R inward + dv/dt tangent
but v = w R, w (omega) in radians/second of arc
so
a = v^2 / R inward + 1.0 tangentially
v = tangential acceleration * t = 1 t
|a| = sqrt ( t^4/R^2 t^2 + 1 ) = 2
so
t^2/R =3
t^2 = 3 * 120
t =19 seconds
w R = v
so
R d w/dt = dv/dt = 1 m/s^2
dw/dt = 1 / 120 radians/second^2
w = dw/dt *t = (1/120)(19) = 0.158 radians /s * 180/3.14) =9.1degrees/sec
theta = (1/2) (1/120)(19^2 ) = 1.5 radians = 86.2 degrees
To determine the angle through which the car will have travelled, we need to use the formulas that relate linear and angular motion in circular motion.
First, let's find the magnitude of the car's tangential acceleration (at).
Given:
Radius of the curve (r) = 120 m
Tangential acceleration (at) = 1.0 m/s²
The tangential acceleration (at) can be defined as at = r * α, where α is the angular acceleration.
Rearranging the equation, we get α = at / r.
Plugging in the given values, α = 1.0 m/s² / 120 m ≈ 0.0083 rad/s².
Next, let's find the magnitude of the car's total acceleration (a).
Given:
Total acceleration (a) = 2.0 m/s²
The total acceleration (a) can be defined as a = √(at² + ar²), where ar is the radial acceleration.
Since the car starts from rest, the radial acceleration (ar) will be equal to v² / r, where v is the linear velocity.
At the start of motion, the linear velocity (v) is zero, so ar = 0.
Therefore, a = √(at² + 0) = at.
Plugging in the given value, a = 2.0 m/s².
Now, let's find the angle through which the car has traveled (θ).
The angular displacement (θ) can be calculated using the equation θ = ω₀t + (1/2)αt², where ω₀ is the initial angular velocity and t is the time.
Since the car starts from rest, ω₀ will be zero.
Using the equation θ = (1/2)αt², we can rearrange it to t = √(2θ / α), solving for t.
Plugging in the known values, t = √(2θ / 0.0083 rad/s²).
Finally, let's find the angle θ.
Rearranging the equation, θ = (0.5) * α * t².
Plugging in the known values, θ = 0.5 * 0.0083 rad/s² * (√(2θ / 0.0083 rad/s²))².
Simplifying the equation, θ = 0.0083 rad/s² * 2θ / 0.0083 rad/s².
Canceling out the terms, 1 = 2θ.
Rearranging the equation, θ = 1 / 2 ≈ 0.5 radians.
Therefore, the car will have traveled approximately 0.5 radians when the magnitude of its total acceleration is 2.0 m/s².