Rotational Kinetics hardest problem

[shaka23h]

Multiple-Concept Example 7 explores the approach taken in problems such as this one. The blades of a ceiling fan have a radius of 0.375 m and are rotating about a fixed axis with an angular velocity of +1.95 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of +2.69 rad/s2. After 0.473 s have elapsed since the switch was reset, what is (a) the total acceleration (in m/s2) of a point on the tip of a blade and (b) the angle between the total acceleration and the centripetal acceleration (See Figure 8.13b)?

I am very lost on this problem.

THe example pretty much just told me that

a = Sqrt ac^2 + aT^2

I have no clue how to begin this problem. please help even if its just a hint.

THanks

 

[OlderDan]

Multiple-Concept Example 7 explores the approach taken in problems such as this one. The blades of a ceiling fan have a radius of 0.375 m and are rotating about a fixed axis with an angular velocity of +1.95 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of +2.69 rad/s2. After 0.473 s have elapsed since the switch was reset, what is (a) the total acceleration (in m/s2) of a point on the tip of a blade and (b) the angle between the total acceleration and the centripetal acceleration (See Figure 8.13b)?

I am very lost on this problem.

THe example pretty much just told me that

a = Sqrt ac^2 + aT^2

I have no clue how to begin this problem. please help even if its just a hint.

THanks

We are going to have to agree on an interpretation of the given information. I assume they mean the the tips of the blades of the fan are .375m from the center of the axis or rotation. Let's just call that R. There is a connection between the angular velocity of the blades, and the speed that a blade tip is moving. Since there is an angular acceleration for some period of time, the angular velocity will be increasing and will reach a new value at the end of the specified time. At that time, the blade tip speed is still increasing, but has a new value that can be calculated from the given information.

At the point in question the blade tip has both a speed and a rate of change of speed. The speed has an associated centripetal acceleration directed toward the center of the circle. The rate of change of speed has an associated acceleration in a direction tangent to the circle. Your job, should you choose to accept it, is to find these two components of acceleration and their resultant and find the angle between the total acceleration and the centripetal component.

 

[kyle8921]

for reaching out for help with this problem. Rotational kinetics can be a challenging concept to grasp, but with some guidance, you can definitely solve this problem.

First, let's break down the information we have been given. The problem is asking us to find the total acceleration of a point on the tip of a blade of a ceiling fan and the angle between the total acceleration and the centripetal acceleration. We are given the radius of the blades (0.375 m), the angular velocity (1.95 rad/s), and the angular acceleration (2.69 rad/s^2).

To solve this problem, we will need to use multiple concepts from rotational kinematics. The first concept we will need is the relationship between angular velocity, angular acceleration, and time:

ωf = ωi + αt

Where ωf is the final angular velocity, ωi is the initial angular velocity, α is the angular acceleration, and t is the time. In this problem, we are given the initial angular velocity (1.95 rad/s), the angular acceleration (2.69 rad/s^2), and the time (0.473 s). We can use this equation to find the final angular velocity.

Next, we will need to use the relationship between linear and angular velocity:

v = rω

Where v is the linear velocity, r is the radius, and ω is the angular velocity. We can use this equation to find the linear velocity of the tip of the blade.

Now, to find the total acceleration, we will use the equation you mentioned:

a = √(ac^2 + aT^2)

Where ac is the centripetal acceleration and aT is the tangential acceleration. We can use the linear velocity we found in the previous step to find the tangential acceleration.

Finally, to find the angle between the total acceleration and the centripetal acceleration, we can use the relationship between these two accelerations and the radius:

tanθ = aT/ac = rω^2/ac

Where θ is the angle between the two accelerations, r is the radius, ω is the angular velocity, and ac is the centripetal acceleration. You can use this equation to find the angle between the two accelerations.

I hope this helps you get started on solving this problem. Remember to use all the relevant concepts and equations to find the solution. If