A car is traveling around a horizontal circular track with radius r = 190.0 m at a constant speed v = 22.0 m/s as shown. The angle èA = 15.0° above the x axis, and the angle èB = 63.0° below the x axis.
1) What is the magnitude of the car’s acceleration?
2) What is the x component of the car’s acceleration when it is at point A
3) What is the y component of the car’s acceleration when it is at point A
4) What is the x component of the car’s acceleration when it is at point B
5) What is the y component of the car’s acceleration when it is at point B
To answer these questions, we need to use the concepts of centripetal acceleration and the components of acceleration.
1) The magnitude of the car's acceleration can be found using the formula for centripetal acceleration:
a = v^2 / r
where v is the velocity and r is the radius of the circular path. Plugging in the values, we get:
a = (22.0 m/s)^2 / 190.0 m ≈ 2.563 m/s^2
So, the magnitude of the car's acceleration is approximately 2.563 m/s^2.
2) The x-component of the car's acceleration at point A can be calculated using the formula:
a_x = a * cos(θ)
where a is the magnitude of acceleration and θ is the angle above the x-axis (15.0°).
Plugging in the values, we get:
a_x = 2.563 m/s^2 * cos(15.0°) ≈ 2.475 m/s^2
So, the x-component of the car's acceleration at point A is approximately 2.475 m/s^2.
3) The y-component of the car's acceleration at point A can be calculated using the formula:
a_y = a * sin(θ)
where a is the magnitude of acceleration and θ is the angle above the x-axis (15.0°).
Plugging in the values, we get:
a_y = 2.563 m/s^2 * sin(15.0°) ≈ 0.663 m/s^2
So, the y-component of the car's acceleration at point A is approximately 0.663 m/s^2.
4) The x-component of the car's acceleration at point B can be calculated using the same formula used for point A:
a_x = a * cos(θ)
where a is the magnitude of acceleration and θ is the angle below the x-axis (63.0°).
Plugging in the values, we get:
a_x = 2.563 m/s^2 * cos(63.0°) ≈ -1.121 m/s^2
Note that the negative sign indicates the direction towards the negative x-axis.
So, the x-component of the car's acceleration at point B is approximately -1.121 m/s^2.
5) The y-component of the car's acceleration at point B can be calculated using the same formula used for point A:
a_y = a * sin(θ)
where a is the magnitude of acceleration and θ is the angle below the x-axis (63.0°).
Plugging in the values, we get:
a_y = 2.563 m/s^2 * sin(63.0°) ≈ -2.342 m/s^2
Note that the negative sign indicates the direction towards the negative y-axis.
So, the y-component of the car's acceleration at point B is approximately -2.342 m/s^2.