The Giant Ferris Wheel has a radius of 30meters. Assuming that a time t= t_0 , the wheel rotates counterclockwise with a speed of 40m/min, and is slowing at a rate of 15m/min^2 . Find acceleration vector for a person seated in a car at the lowest point of the wheel.
How to find acceleration vector?
And i don't get what does it meant by acceleration vector is.
"confuse"
Please give a hand. Thank You.
To find the acceleration vector for a person seated in a car at the lowest point of the wheel, we need to understand what an acceleration vector is and how to calculate it.
An acceleration vector represents the rate of change of velocity of an object along with its direction. It is a vector quantity because it has both magnitude (the amount of acceleration) and direction (the direction in which the acceleration is occurring).
In the case of the person seated in a car at the lowest point of the wheel, there are two types of acceleration to consider: tangential acceleration and centripetal acceleration.
Tangential acceleration is the component of acceleration in the tangential direction (along the circumference of the wheel). It accounts for the change in the speed of the wheel due to its slowing down. In this case, the tangential acceleration can be calculated using the formula:
a_t = dv/dt
where:
a_t is the tangential acceleration,
dv is the change in velocity, and
dt is the change in time.
Given that the speed of the wheel at time t= t_0 is 40 m/min and it is slowing down at a rate of 15 m/min^2, we can compute the tangential acceleration at time t= t_0.
a_t = -15 m/min^2
The negative sign indicates that the wheel is slowing down.
Centripetal acceleration is the component of acceleration towards the center of the wheel. It is responsible for keeping the person seated in the car moving in a circular path. For a object moving in a circle, the centripetal acceleration can be calculated using the formula:
a_c = v^2 / r
where:
a_c is the centripetal acceleration,
v is the tangential velocity (speed), and
r is the radius of the circle (in this case, the radius of the wheel).
At the lowest point of the wheel, the tangential velocity is the same as the speed of the wheel because the wheel is rotating counterclockwise. So, the tangential velocity (v) is 40 m/min. And the radius (r) of the wheel is given as 30 meters.
a_c = (40 m/min)^2 / 30 m
Now, we can calculate the centripetal acceleration at the lowest point of the wheel.
a_c = 53.33 m/min^2
Since the centripetal acceleration acts towards the center of the wheel, its direction is towards the center (towards the axis of rotation).
To get the total acceleration vector, we combine the tangential acceleration and centripetal acceleration vectors.
The tangential acceleration vector (a_t) acts tangentially to the circumference (along the wheel's motion), and the centripetal acceleration vector (a_c) acts towards the center of the wheel. The total acceleration vector is the vector sum of these two vectors.
To find the total acceleration vector, we can calculate the magnitudes and combine them using vector addition.
The magnitude of the total acceleration will be:
|a_total| = √(a_t^2 + a_c^2)
Then, to find the direction, we can calculate the angle between the tangential acceleration and centripetal acceleration vectors:
θ = tan^(-1)(a_c/a_t)
Finally, the total acceleration vector can be given as:
a_total = |a_total| * cos(θ) * i + |a_total| * sin(θ) * j
where i and j are the unit vectors representing the x-axis and y-axis directions, respectively.
By plugging in the values calculated above, you can obtain the acceleration vector for a person seated in a car at the lowest point of the wheel.