A 41.0 cm diameter wheel accelerates uniformly from 86.0 rpm to 342.0 rpm in 3.4 s. How far (in meters) will a point on the edge of the wheel have traveled in this time?
Linear speed=(RPM/60)xpixD, t=3.4s, D=0.41m
u=86x0.41x3.14/60=1.85m/s
v=342x3.41x0.41/60=7.34m/s
v=u+ft or f=(7.34-1.85)/3.4=1.61m/s^2
s=ut+ft^2/2=1.85x3.4 + 1.61x3.4^2/2 =6.29+9.31=15.60m
To find the distance traveled by a point on the edge of the wheel, we need to first find the angular acceleration and then use it to calculate the angular displacement. Once we have the angular displacement, we can convert it to linear displacement using the formula:
Distance = (angular displacement) * (radius of the wheel)
Given:
Diameter = 41.0 cm
Radius = Diameter/2 = 41.0 cm / 2 = 20.5 cm = 0.205 m
Initial angular speed = 86.0 rpm
Final angular speed = 342.0 rpm
Time = 3.4 s
Step 1: Convert the initial and final angular speeds from rpm to radians per second.
1 rpm = (2π/60) rad/s
Initial angular speed = 86.0 rpm * (2π/60) rad/s = 9.04 rad/s
Final angular speed = 342.0 rpm * (2π/60) rad/s = 35.92 rad/s
Step 2: Calculate the angular acceleration.
Angular acceleration = (Change in angular speed) / Time
Angular acceleration = (Final angular speed - Initial angular speed) / Time
Angular acceleration = (35.92 rad/s - 9.04 rad/s) / 3.4 s
Angular acceleration = 7.68 rad/s²
Step 3: Calculate the angular displacement.
Angular displacement = (Initial angular speed * Time) + (0.5 * Angular acceleration * Time²)
Angular displacement = (9.04 rad/s * 3.4 s) + (0.5 * 7.68 rad/s² * (3.4 s)²)
Angular displacement = 30.736 rad
Step 4: Calculate the linear displacement.
Distance = Angular displacement * Radius
Distance = 30.736 rad * 0.205 m
Distance ≈ 6.30 m
Therefore, a point on the edge of the wheel will have traveled approximately 6.30 meters in 3.4 seconds.