Calculating Angular Velocity of Double Gear and Velocities of Upper Rack and Gear Point D
The double gear rolls on the stationary lower rack; the velocity of its center is 1.2 m/s. Determine (a) the angular velocity of the gear, and (b) the velocities of the upper rack R (point B) and point D of the gear.
Double gear roll, stationary lower rack, 1.2 m/s, and angular velocity.
(a) Angular Velocity of the Gear
To determine the angular velocity of the gear, first we need to find the gear's radius. The given dimensions are 100 mm and 150 mm. The total diameter of the gear is 100 mm + 150 mm = 250 mm. Therefore, the radius is 125 mm / 2 = 125 mm. Next, we need to convert the radius from millimeters to meters: 125 mm * (1 m / 1000 mm) = 0.125 m. Now we can use the formula for linear velocity (v) and angular velocity (ω): v = rω, where r is the radius. We can rearrange the formula to find ω: ω = v / r. Plug in the given velocity, 1.2 m/s, and calculated radius, 0.125 m: ω = 1.2 m/s / 0.125 m = 9.6 rad/s.
(a) The angular velocity of the gear is 9.6 rad/s.
(b) Velocities of Upper Rack R (Point B) and Point D of the Gear
To find the velocities of the upper rack R (point B) and point D of the gear, we'll use the formula v = rω. For point B, the radius is 150 mm or 0.150 m. Using the angular velocity from part (a), we have v_B = 0.150 m × 9.6 rad/s = 1.44 m/s. For point D, the radius is 100 mm or 0.100 m. Similarly, we have v_D = 0.100 m × 9.6 rad/s = 0.96 m/s.
(b) The velocities of the upper rack R (point B) and point D of the gear are 1.44 m/s and 0.96 m/s, respectively.
Question:
What is the method used to determine the angular velocity of the gear and calculate the velocities of the upper rack and gear point D?
Answer:
The method used to determine the angular velocity of the gear involved finding the gear's radius, converting it to meters, and using the formula ω = v / r. To calculate the velocities of the upper rack R (point B) and point D of the gear, the formula v = rω was utilized with the respective radii of 150 mm and 100 mm.