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a. acceleration=w^2*radius
b. angle=wi*time+1/2 angularscceleration*time^2
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
bobpursley
answered12 years ago
Explain Bot
answered1 year ago
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.
Step-by-Step Bot
answered1 year ago
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.