Question

1. The angular velocity of a fan blade is 8.4 radians per second.

Answer 1

Answer:

1. The fan blade makes 112.3 revolutions in 1.4 minutes

2. The linear velocity of the gears is the same. The linear velocity is 432π centimeters per minute ⇒ B

Step-by-step explanation:

1.

∵ The angular velocity of a fan blade is 8.4 radians per second

∴ ω = 8.4 rad/sec

- The revolution = the product of angular velocity and the time

   of revolution ÷ 2π

∵ Revolutions = (ω . t)/2π

∵ The fan blade makes x revolutions in 1.4 minutes

∴ t = 1.4 minutes

- Change it to seconds

∵ 1 minute = 60 seconds

∴ 1.4 minutes = 1.4 × 60 = 84 seconds

- Substitute ω and t in the rule of revolutions

∴ Revolutions = $\frac{(8.4)(84)}{2\pi }$

∴ Revolutions ≅ 112.2997278

∴ Revolutions ≅ 112.3

- Round it to the nearest tenth

∴ The fan blade makes 112.3 revolutions in 1.4 minutes

2.

∵ The radius of the larger gear is 8 centimeters

∵ The radius of the small gear is 3 centimeters

∵ The two gears are connected by a chain

- That means the move the same distance together

∴ $n_{s}.C_{s}=n_{l}.C_{l}$ , where n is the number of revolution for each

   gear and C is the circumference of each gear

The formula of the circumference of a circle is C = 2πr

∵ $C_{s}$ = 2π(3) = 6π

∵ $C_{l}$ = 2π(8) = 16π

∵ The smaller gear completes 24 revolutions in 20 seconds

∴ $n_{s}$ = 24

- Substitute them in the rule above to find $n_{l}$

∵ 24 × 6π = $n_{l}$ × 16π

∴ 144π = $n_{l}$ × 16π

- Divide both sides by 16π

∴ 9 =  $n_{l}$

∴ The larger gear completes 9 revolutions in 20 seconds

The linear velocities of the two gears are equal because they move together and make the same distance in the same time

∵ The linear velocity = Distance ÷ Time

∵ Distance which each gear makes = C × n

∴ V = $\frac{nC}{t}$

Let us use the smaller gear to find the linear velocity

∵ C = 6π centimeters

∵ n = 24 revolution

∵ t = 20 seconds

- Change it to minutes

∴ 20 seconds = 20 ÷ 60 = $\frac{1}{3}$ minute

∴ V = $\frac{(24)(6\pi )}{\frac{1}{3}}$

∴ V = 432π cm/min

Let us check the answer by find the linear velocity fro the larger gear

∵ C = 16π centimeters

∵ n = 9 revolution

∵ t = $\frac{1}{3}$ minute

∴ V = $\frac{(9)(16\pi )}{\frac{1}{3}}$

∴ V = 432π cm/min

The linear velocity of the gears is the same. The linear velocity is 432π centimeters per minute