Hi!could anyone help me please, I don't get it. Thank you so much for your help.
Question:(a) What is the tangential acceleration of a bug on the rim of a 9.0 in. diameter disk if the disk moves from rest to an angular speed of 79 rev/min in 4.0 s?
___m/s2
(b) When the disk is at its final speed, what is the tangential velocity of the bug?
___ m/s
(c) One second after the bug starts from rest, what is its tangential acceleration?
___ m/s2
What is its centripetal acceleration?
___ m/s2
What is its total acceleration?
___m/s2
___° (relative to the tangential acceleration)
(a) First of all, convert 79 rpm to radians per second, which we will call w.
w = 79 rev/min* (2 pi rad/rev) * (1 min/60s) = 8.27 rad/s
Divide that final angular velocity by 4 seconds to obtain the angular acceleration. Call that number "alpha"
alpha = (8.27 rad/s)/4.0s = 2.07 rad/s^2
The tangential acceleration of the bug
while the disk is speeding up is
a_t = alpha*R
where R = 4.5 in/39.37 in/meter = 0.1143 m. Therefore
a_t = 0.236 m/s^2
b) The tangential velocity at full speed is v_t = R*w
c) same tangential acceleration as (a)
centripetal acceleration @ t=1 s is
a_c (@ t=1) = R w^2, where w is 1/4 of the value calculated in (a), since you are only 1/4 way through the acceleration interval.
For the total acceleration at t=1 s, use
a = sqrt[a_t^2 + a_c^2]
The tangent of the total acceleration vector relative to tangential direction is (a_c)/(a_t)
Thise are the formulas to use. You do the numbers and try to undertand where the formulas came from. Check my work as well. No guarantees.
To find the answers to these questions, we need to use some equations relating to rotational motion. The first thing we need to do is convert the given diameter to its corresponding radius.
Given:
Diameter of the disk = 9.0 in.
(a) To find the tangential acceleration of the bug on the rim of the disk, we can use the equation:
tangential acceleration = (change in angular velocity) x (radius of the disk)
The change in angular velocity can be calculated by converting the final angular speed from rev/min to rad/s, and dividing it by the time taken to reach that speed.
We can use the conversion factor: 2π radians = 1 revolution.
Step 1: Convert the angular speed from rev/min to rad/s.
Final angular speed = 79 rev/min
= (79 rev/min) x (2π radians/1 rev) x (1 min/60 s)
= 79 x 2π/60 rad/s
Step 2: Calculate the change in angular velocity.
Change in angular velocity = (final angular velocity - initial angular velocity)
Since the disk starts from rest, the initial angular velocity is 0 rad/s.
Change in angular velocity = (79 x 2π/60) rad/s - 0 rad/s
Next, we need to convert the diameter given in inches to meters since we want the answer in m/s^2.
Step 3: Convert the diameter to radius.
Radius of the disk = (9.0 in./2) x (2.54 cm/in) x (1 m/100 cm)
= (9.0/2) x (0.0254) m
Step 4: Substitute the values into the equation for tangential acceleration:
tangential acceleration = (change in angular velocity) x (radius of the disk)
(b) To find the tangential velocity of the bug when the disk is at its final speed, we can use the formula:
tangential velocity = (angular velocity) x (radius of the disk)
We can use the final angular velocity calculated in part (a) and the radius of the disk.
(c) To find the tangential acceleration one second after the bug starts from rest, we need to use a different formula for tangential acceleration:
tangential acceleration = (2π x angular velocity) / time
We can use the given time of 1 second and the initial angular velocity as 0 rad/s.
For the centripetal acceleration, it is given by the formula:
centripetal acceleration = (angular velocity)^2 x (radius of the disk)
The total acceleration is the vector sum of the tangential and centripetal accelerations.
I hope this explanation helps you to solve the problem! If you have any further questions, feel free to ask.