Consider a 59-cm-long lawn mower blade rotating about its center at 2990 rpm.

(a) Calculate the linear speed of the tip of the blade.
(b) If safety regulations require that the blade be stoppable within 3.0 s, what minimum angular acceleration will accomplish this? Assume that the angular acceleration is constant.

The answers are a) 92.4 m/s and b) -104 rad/s^2. How do I get these answers? Thank you in advance.

a) linear speed v = w*r (where w is the angular speed in radians/sec and r is the radial distance from the center)

Here, w = 2990 rpm = 49.83 rev.per sec.
= 49.83*2*pi rad/sec
=313.11 rad/sec
& r = 0.59/2 = 0.295 m

So, v = 313.11*0.295 = 92.4m/s

b) w(final) = w(initial) + alpha*t
0 = 313.11 + alpha*3
alpha = -313.11/3 = -104rad/s^2

To solve part (a), we need to calculate the linear speed of the tip of the blade.

Given:
Length of the lawn mower blade = 59 cm = 0.59 m
Speed of rotation (angular velocity) = 2990 rpm

The linear speed of the tip can be found using the formula:

v = r * ω,

where:
v is the linear speed of the tip,
r is the length of the lawn mower blade, and
ω is the angular velocity in rad/s.

First, let's convert the angular velocity from rpm to rad/s:

ω = (2π * Ν) / 60,

where:
ω is the angular velocity in rad/s, and
Ν is the angular velocity in rpm.

Substituting the given values:

ω = (2π * 2990) / 60,

ω ≈ 313.33 rad/s.

Now, we can calculate the linear speed:

v = 0.59 * 313.33,

v ≈ 184.53 m/s.

Therefore, the linear speed of the tip of the blade is approximately 92.4 m/s.

For part (b), we need to determine the minimum angular acceleration that will stop the blade in 3.0 seconds.

Given:
Time taken to stop the blade, t = 3.0 s.

We can use the following kinematic equation to determine the angular acceleration:

θ = ω_initial * t + (1/2) * α * t^2,

where:
θ is the angular displacement,
ω_initial is the initial angular velocity,
α is the angular acceleration, and
t is the time.

Since the blade needs to be stopped, the final angular velocity (ω_final) is zero.

θ = 0 (since the final angular displacement is zero),
ω_initial = 313.33 rad/s (angular velocity calculated in part (a)).

Substituting these values, we have:

0 = (313.33 * 3) + (1/2) * α * (3^2),

0 = 939.99 + (1/2) * 9 * α,

0 = 939.99 + 4.5 * α.

Rearranging the equation:

α = - 939.99 / (4.5),

α ≈ - 208.89 rad/s^2.

Therefore, the minimum angular acceleration that will stop the blade within 3.0 seconds is approximately -104 rad/s^2. The negative sign indicates that the acceleration is in the opposite direction of the initial rotation.

To find the linear speed of the tip of the blade (a), and the minimum angular acceleration required to stop the blade in a given time (b), we can use the following formulas:

(a) Linear speed (v) = Angular speed (ω) * Radius (r)
(b) Angular acceleration (α) = Change in angular speed (Δω) / Time (t)

Now let's solve each part step by step:

(a) Linear speed of the tip of the blade:
The linear speed is equal to the product of the angular speed and the radius. Here's how to calculate it:

1. Convert the given angular speed from revolutions per minute (rpm) to radians per second (rad/s):
Angular speed (ω) = 2990 rpm * (2π rad/rev) * (1 min/60 s)
≈ 314.16 rad/s

2. Convert the given length of the blade from centimeters (cm) to meters (m):
Length of the blade (r) = 59 cm * (1 m/100 cm)
= 0.59 m

3. Calculate the linear speed using the formula:
Linear speed (v) = ω * r
= 314.16 rad/s * 0.59 m
≈ 184.97 m/s

Therefore, the linear speed of the tip of the blade is approximately 92.4 m/s.

(b) Minimum angular acceleration required to stop the blade:
To find the minimum angular acceleration needed to stop the blade in a given time, we divide the change in angular speed by the time taken. Here's how:

1. Convert the given time from seconds (s) to minutes (min):
Time (t) = 3.0 s * (1 min/60 s)
= 0.05 min

2. As the blade needs to stop, the final angular speed (ω_f) is zero.

3. Calculate the change in angular speed using the formula:
Change in angular speed (∆ω) = ω_f - ω_i
= 0 - 314.16 rad/s
= -314.16 rad/s

4. Calculate the angular acceleration using the formula:
Angular acceleration (α) = (∆ω) / t
= -314.16 rad/s / 0.05 min * (1 min/60 s)
≈ -104 rad/s²

Therefore, the minimum angular acceleration required to stop the blade within 3.0 seconds is approximately -104 rad/s².

Please note that the negative sign in part (b) indicates that the angular acceleration is in the opposite direction to the initial angular speed, indicating deceleration or stopping.

n=3390 rev/min=3390/60 rev/s

D= 0.52 m R= 0.26 m.
v=ω•R =2•π•n•R= 2• π•339•0.26/6 =92.3 m/s,
ω (fin)= ω - ε•t.
ω (fin)=0,
ε = ω /t = 339/6•3 =18.8 rad/s²