A racing car, starting from rest, travels around a circular turn of radius 28.9 m. At a certain instant, the car is still accelerating, and its angular speed is 0.421 rad/s. At this time, the total acceleration (centripetal plus tangential) makes an angle of 35.0° with respect to the radius. What is the magnitude of the total acceleration?
To find the magnitude of the total acceleration, we need to find the components of the acceleration in the radial and tangential directions.
Given:
Radius of the circular turn (r) = 28.9 m
Angular speed (ω) = 0.421 rad/s
Angle between acceleration and radius (θ) = 35.0°
Step 1: Convert the angle from degrees to radians
θ = 35.0° × (π/180) ≈ 0.6109 rad
Step 2: Find the tangential acceleration (at)
The tangential acceleration can be calculated using the formula
at = r × α
where α is the angular acceleration.
Since the car starts from rest, it has no initial angular velocity, so α = ω/t
where t is the time in seconds.
We need to calculate t:
ω = αt
t = ω/α
Before calculating t, we need to find α.
α = (vf - vi)/t
where vf is the final angular velocity, vi is the initial angular velocity, and t is the time.
Since the car starts from rest, vi = 0.
α = vf/t
vf = ω (given)
α = ω/t
t = ω/α
t = ω/(ω/t)
t = ω²/α
Therefore, t = (0.421 rad/s)²/α
Step 3: Find α:
vf = ω = 0.421 rad/s
vi = 0 rad/s
t = t (calculated from Step 2)
α = (vf - vi)/t
α = (0.421 rad/s - 0 rad/s)/t
Step 4: Calculate α:
α = 0.421 rad/s / t
Substitute the calculated value of t from Step 2 and solve for α.
Step 5: Calculate at:
at = r × α
Substitute the value of α calculated in Step 4 and the given value of r.
Step 6: Calculate ac (centripetal acceleration):
ac = (r × ω²)
Substitute the given values of r and ω.
Step 7: Find the components of the total acceleration:
The total acceleration can be divided into two components: the centripetal acceleration (ac) and the tangential acceleration (at).
To find the magnitude of the total acceleration, we can use the Pythagorean theorem:
|a|² = |ac|² + |at|²
Substitute the calculated values of ac and at, and solve for |a|.
Step 8: Calculate the magnitude of the total acceleration:
|a| = √(|ac|² + |at|²)
Substitute the calculated values of |ac| and |at|, and solve for |a|.
To find the magnitude of the total acceleration, we need to understand the components of the total acceleration.
The total acceleration is made up of two components: the centripetal acceleration (ac) and the tangential acceleration (at).
Centripetal acceleration is the acceleration towards the center of the circular path, and it can be calculated using the formula:
ac = ω² * r
where ω is the angular speed and r is the radius of the circular path.
Tangential acceleration is the acceleration in the direction of motion along the circular path. It can be calculated using the formula:
at = α * r
where α is the angular acceleration and r is the radius of the circular path.
However, in this case, we are not given the angular acceleration. Instead, we are given the angular speed (ω) at a certain instant and the angle between the total acceleration and the radius (θ). By using this information, we can calculate the tangential acceleration.
The tangential acceleration can be found using the formula:
at = ω² * r * tan(θ)
Once we have both the centripetal and tangential accelerations, we can find the magnitude of the total acceleration (a) using the Pythagorean theorem:
a = √(ac² + at²)
Let's calculate the magnitude of the total acceleration using the given values:
Given:
Radius (r) = 28.9 m
Angular speed (ω) = 0.421 rad/s
Angle (θ) = 35.0°
First, convert the angle from degrees to radians:
θ = 35.0° * (π/180) ≈ 0.611 rad
Next, calculate the tangential acceleration:
at = ω² * r * tan(θ)
= (0.421 rad/s)² * (28.9 m) * tan(0.611 rad)
≈ 4.89 m/s²
Now, calculate the centripetal acceleration:
ac = ω² * r
= (0.421 rad/s)² * (28.9 m)
≈ 5.4 m/s²
Finally, calculate the magnitude of the total acceleration:
a = √(ac² + at²)
= √((5.4 m/s²)² + (4.89 m/s²)²)
≈ √(29.16 m²/s⁴ + 23.88 m²/s⁴)
≈ √(53.04 m²/s⁴)
≈ 7.28 m/s²
Therefore, the magnitude of the total acceleration is approximately 7.28 m/s².