A car starts a race from rest on a circular track and has a tangential speed of 43 m/s at the end of the third lap. The track has a radius of 91 m. If it has constant angular acceleration, what is the magnitude of its tangential acceleration?
Help me and please explain in simple terms because I'm very confused.
a(τ) =ε•R =(v²/2•φ) •R =v²•R/2 • 2•π•N = v²•R/4•π•N
what is N
To find the magnitude of the car's tangential acceleration, we need to understand a few concepts and use some formulas.
First, let's understand some key terms:
1. Tangential speed: It is the linear speed of an object moving along a circular path. In this case, the tangential speed is given as 43 m/s.
2. Radius: It is the distance between the center of the circular path and any point on the circumference. Here, the radius of the track is 91 m.
3. Angular acceleration: It is the rate at which an object's angular velocity (how fast it's rotating) changes over time. In this problem, we are told that the car has constant angular acceleration, meaning it's rotating at a constant rate.
Now, let's find the magnitude of the car's tangential acceleration step by step:
Step 1: Calculate the car's angular velocity (ω) at the end of the third lap.
The formula to calculate angular velocity is given by:
ω = v / r
where v is the tangential speed and r is the radius.
Plugging in the values:
ω = 43 m/s / 91 m
Step 2: Calculate the time taken for the car to complete three laps.
The time taken for the car to complete one lap is equal to the time taken for one complete rotation. Since we are given that the car has constant angular acceleration, we can use the formula:
θ = ω_initial * t + (1/2) * α * t^2
where θ is the angle of rotation, ω_initial is the initial angular velocity, α is the angular acceleration, and t is the time taken.
For one lap, θ = 2π radians (since it's a complete circle).
ω_initial is 0 since the car starts from rest.
Substituting these values, we get:
2π = (1/2) * α * t^2
To find the time taken for three laps, we multiply the time taken for one lap by three:
3 * 2π = (1/2) * α * t_total^2
Step 3: Calculate the total time taken (t_total).
Simplifying the equation we got in Step 2, we have:
6π = α * t_total^2
Step 4: Calculate the angular acceleration (α).
Rearranging the equation from Step 3, we can solve for α:
α = (6π) / t_total^2
Step 5: Calculate the tangential acceleration (a_tangential).
The formula for the tangential acceleration is:
a_tangential = r * α
Plugging in the values:
a_tangential = 91 m * [(6π) / t_total^2]
By following these steps, you can find the magnitude of the car's tangential acceleration. Remember to calculate the total time taken for three laps (Step 3) before calculating the angular acceleration (Step 4) and finally the tangential acceleration (Step 5).