The blades of a ceiling fan have a radius of 0.348 m and are rotating about a fixed axis with an angular velocity of +1.63 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of 2.16 rad/s^2. After 0.545 s have elapsed since the switch was reset, what are the following?

Question

The blades of a ceiling fan have a radius of 0.348 m and are rotating about a fixed axis with an angular velocity of +1.63 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of 2.16 rad/s^2. After 0.545 s have elapsed since the switch was reset, what are the following?

(a) the total acceleration (in m/s^2) of a point on the tip of a blade
(b) the angle between the total acceleration and the centripetal acceleration

Answer 1

To solve this problem, we need to use the equations of rotational motion. Let's break it down step by step:

(a) The total acceleration of a point on the tip of a blade can be calculated using the formula:

acceleration = radius * angular acceleration + (angular velocity)^2 * radius

Given:
radius (r) = 0.348 m
angular acceleration (α) = 2.16 rad/s^2
angular velocity (ω) = 1.63 rad/s

Substituting the values into the formula:

acceleration = (0.348 m) * (2.16 rad/s^2) + (1.63 rad/s)^2 * (0.348 m)

Now, let's perform the calculation:

acceleration = 0.751 m/s^2

So, the total acceleration of a point on the tip of a blade is 0.751 m/s^2.

(b) The angle (θ) between the total acceleration and the centripetal acceleration can be calculated using the formula:

cos(θ) = centripetal acceleration / total acceleration

centripetal acceleration = (angular velocity)^2 * radius

Given:
radius (r) = 0.348 m
angular velocity (ω) = 1.63 rad/s

Substituting the values into the formula:

centripetal acceleration = (1.63 rad/s)^2 * (0.348 m)

Now, let's perform the calculation:

centripetal acceleration = 0.951 m/s^2

Now, let's calculate the angle using the formula:

cos(θ) = (0.951 m/s^2) / (0.751 m/s^2)

Now, find the angle (θ) by taking the inverse cosine (cos^-1) of both sides:

θ = cos^-1(0.951 m/s^2 / 0.751 m/s^2)

Now, let's calculate the angle using a calculator:

θ ≈ 22.51 degrees

So, the angle between the total acceleration and the centripetal acceleration is approximately 22.51 degrees.

Answer 2

To answer the given questions, we need to use some key formulas related to circular motion.

For a point rotating in a circle with radius r and angular velocity ω, the linear velocity (v) is given by:
v = r * ω

The centripetal acceleration (ac) of a point rotating in a circle with radius r and linear velocity v is given by:
ac = v^2 / r

The total acceleration (at) of a point rotating in a circle with radius r and angular velocity ω, as well as angular acceleration α, is given by:
at = √(ac^2 + (r * α)^2)

Let's calculate the answers step by step:

Step 1: Calculate the linear velocity (v) of the point on the tip of the blade.
v = r * ω
= 0.348 m * 1.63 rad/s

Step 2: Calculate the centripetal acceleration (ac) of the point.
ac = v^2 / r
= (v * v) / r

Step 3: Calculate the total acceleration (at) of the point.
at = √(ac^2 + (r * α)^2)
= √((v^2 / r)^2 + (0.348 m * 2.16 rad/s^2)^2)

Step 4: Calculate the angle between the total acceleration (at) and the centripetal acceleration (ac).
To find the angle, we can use the formula:
tanθ = (r * α) / ac

Now we have the values for all the variables, we can substitute them into the formulas and calculate the answers.

Step 1: Calculate the linear velocity (v) of the point on the tip of the blade.
v = 0.348 m * 1.63 rad/s
= 0.567 m/s

Step 2: Calculate the centripetal acceleration (ac) of the point.
ac = (v * v) / r
= (0.567 m/s)^2 / 0.348 m
= 0.924 m/s^2

Step 3: Calculate the total acceleration (at) of the point.
at = √((v^2 / r)^2 + (0.348 m * 2.16 rad/s^2)^2)
= √((0.567 m/s)^4 / (0.348 m)^2 + (0.348 m * 2.16 rad/s^2)^2)
= √(0.608 m^2/s^4 + 0.258 m^2/s^4)
= √0.866 m^2/s^4
≈ 0.93 m/s^2

Step 4: Calculate the angle between the total acceleration (at) and the centripetal acceleration (ac).
tanθ = (r * α) / ac
θ = atan((r * α) / ac)
= atan(0.348 m * 2.16 rad/s^2 / 0.924 m/s^2)

Now, substitute the values into the equation and calculate the angle.

θ = atan(0.348 m * 2.16 rad/s^2 / 0.924 m/s^2)
≈ 1.00 radians

So, the answers to the given questions are:
(a) The total acceleration of a point on the tip of a blade is approximately 0.93 m/s^2.
(b) The angle between the total acceleration and the centripetal acceleration is approximately 1.00 radians.

Answer 3

a. acceleration=w^2*radius

b. angle=wi*time+1/2 angularscceleration*time^2