Rotational Mechanics and Machines

By conservation of angular momentum, I1 * w1 = I2 * w2. Given I2 = 0.6 * I1 and w1 = 120 rpm, we have 120 * I1 = w2 * 0.6 * I1, which simplifies to w2 = 120 / 0.6 = 200 rpm.

Mechanical advantage is the amplification in force due to the use of a machine . The mechanical advantage of wheel and axle is given by 

The mechanical advantage of a wheel and axle $= \displaystyle \frac{Radius \ of \ wheel}{Radius \ of \ the \ axle}$
$MA _A = \displaystyle \frac{R _1}{r _1} $ and  $MA _B = \displaystyle \frac{R _2}{r _2}$
Given $ r _1  :  r _2  = 5 : 4$
$R _1  :  R _2  = 3  :  7$
$ \displaystyle \frac{MA _A}{MA _B} = \frac{R _1}{r _1} \times \frac{r _2}{R _2}$
$\displaystyle = \frac{3}{5} \times \frac{4}{7} = \frac{12}{35}$
$MA _B >  MA _A$

The mechanical advantage of a wheel and axle $= \displaystyle \frac{Radius  of  wheel}{Radius  of  the  axle}$
Given radius of wheel $=$ 25 $\times$ radius of axle
$MA = \displaystyle \frac{25 \times radius  of  axle}{radius  of  axle} = 25$

The frequency (f) of a wave is the number of full wave forms generated per second. This is the same as the number of repetitions per second or the number of oscillations per second.  
Time Period (T) is the number of seconds per waveform, or the number of seconds per oscillation.  It is clear that frequency and period are reciprocals. 
In this case, A bicycle wheel spins 25 times in 5 seconds. 
That is, the frequency is given as $f=\frac { 25 }{ 5 } =5Hz$.
Time period = $\frac { 1 }{ f } =\frac { 1 }{ 5 } =0.20s$
Hence, the frequency is 5 Hz and the time period is 0.20 seconds.

A wheel and axle is a simple machine where a larger wheel is attached to a smaller axle. Because the wheel has a greater radius, it allows for mechanical advantage.

At the highest point of the Ferris wheel, the angle between the horizontal and the line connecting the passenger to the center of the wheel is 90 degrees.

For recreational cycling, a cadence of 80-90 revolutions per minute (rpm) is generally recommended to maintain a balance between efficiency and comfort.

The circumference of the Ferris wheel is π * 50 meters = 157.08 meters. Since the Ferris wheel makes one complete revolution in 10 minutes, it travels 157.08 meters in 10 minutes. Therefore, in 5 minutes, it travels half of that distance, which is 78.54 meters. Since the person boards the Ferris wheel at the bottom, they will be 78.54 meters above the ground after 5 minutes.

The circumference of the Ferris wheel is π * 40 meters = 125.66 meters. Since the Ferris wheel makes one complete revolution in 8 minutes, it travels 125.66 meters in 8 minutes. Therefore, in 4 minutes, it travels half of that distance, which is 62.83 meters. Since the person boards the Ferris wheel at the bottom, they will be 62.83 meters above the ground after 4 minutes.

The circumference of the Ferris wheel is π * 30 meters = 94.25 meters. Since the Ferris wheel makes one complete revolution in 6 minutes, it travels 94.25 meters in 6 minutes. Therefore, in 3 minutes, it travels half of that distance, which is 47.13 meters. Since the person boards the Ferris wheel at the bottom, they will be 47.13 meters above the ground after 3 minutes.

Extreme mountain bikes commonly use wheel sizes of 27.5 inches or 29 inches, which provide a good balance of stability and agility on challenging trails.

A cadence of 80-90 rpm is generally recommended for extreme biking on technical trails, as it provides a balance between power and control.

The recommended spoke count for a road bike wheel is typically 32 spokes. This number of spokes provides a good balance between strength, weight, and durability for road cycling.

The rolling resistance of a vehicle is calculated by multiplying the coefficient of rolling resistance between the vehicle's tires and the road by the mass of the vehicle and the acceleration due to gravity.