a. acceleration=w^2*radius
b. angle=wi*time+1/2 angularscceleration*time^2
(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
b. angle=wi*time+1/2 angularscceleration*time^2
(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.
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.