Question

Modern wind turbines are larger than they appear, and despite their apparently lazy motion, the speed of the blades tips can be quite high-many times higher than the wind speed. A turbine has blades 52 m long that spin at 12 rpm .
At the tip of a blade, what are (a) the speed and (b) the centripetal acceleration?

Answer 1

Answer:

(a) v = 65.35 m/s

(b) ac = 82.16 m/s²

Explanation:

Kinematic of the blades of the wind turbine

The blades of the wind turbine describe circular motion and the formulas that apply to this movement are as follows:

v = ω * R   Formula (1)

Where:

v : tangential velocity (m/s)

ω : angular velocity (rad/s)

R : radius of the particle path (m)

The velocity vector is tangent at each point to the trajectory and its direction is that of movement. This implies that the movement has centripetal acceleration (ac):

ac =  ω²* R Formula (1)

ac : centripetal acceleration (m/s²)

Data:

ω=  12 rpm = 12 rev/min

1 rev = 2π rad

1 min = 60 s

ω=  12 rev/min = 12 (2π rad)/(60 s)

ω = 1.257 rad/s

R = 52 m

(a)Tangential velocity at the tip of a blade (v)

We apply the formula (1)

v =  ω* R

v =  ( 1.257)* (52) = 65.35 m/s

(a)  Centripetal acceleration at the tip of a blade (ac)

We apply the formula (2)

ac =  ω²*R

ac =   ( 1.257)²* (52) = 82.16 m/s²

Answer 2

This question involves the concepts of linear velocity and centripetal acceleration.

(a) Linear velocity at the tip of the blade is "65.34 m/s".

(b) The centripetal acceleration is found to be "82.11 m/s²".

(a)

The linear velocity at the tip of the blade is given by the formula:

$v=r\omega$

where,

ω = Angular Speed = 12 rpm$(\frac{1\ min}{60\ s})(\frac{2\pi\ rad}{1\ rev})$ = 1.26 rad/s

v = linear speed at the tip of blade = ?

r = length of blade = 52 m

Therefore,

$v=(52\ m)(1.26\ rad/s)$

v = 65.34 m/s

(b)

The centripetal acceleration can be found using the following formula:

$a_c=\frac{v^2}{r}\\\\a_c=\frac{(65.34\ m/s)^2}{52\ m}\\\\a_c=82.11\ m/s^2$

Learn more about the centripetal acceleration here:

https://brainly.com/question/14465119?referrer=searchResults

The attached picture explains the concept of centripetal acceleration.