9.2.11 In a mechanism with internal gearing, gear 1 and crank OA rotate independently of each other with angular velocities ?1 = 2 rad/s and ?OA = 4 rad/s. It is necessary to determine the angular velocity of gear 2 if the radius r1 = 30 cm and the length of the crank OA is 20 cm.
To solve this problem, you need to use the formula for the speed of a point on a circle: v = r * w, where v is the speed of a point on a circle, r is the radius of the circle, w is the angular speed of rotation of the circle.
First, let's determine the angular velocity of point A located on the crank. To do this, we use the formula to determine the angular velocity of a point on the curve: w_A = w_OA * cos(alpha), where w_OA is the angular velocity of rotation of the crank, alpha is the angle between the crank and the line passing through the center of rotation of the crank and point A. Since the crank rotates with an angular velocity of 4 rad/s, and the angle alpha is 90 degrees, then the angular velocity of point A will be equal to w_A = 4 * cos(90) = 0 rad/s.
Then we determine the speed of point B located on gear 1. To do this, we use the formula v_B = r1 * w1, where r1 is the radius of gear 1, w1 is the angular speed of rotation of gear 1. Since gear 1 rotates with an angular speed of 2 rad /s, then the speed of point B will be equal to v_B = 30 * 2 = 60 cm/s.
Finally, we determine the angular velocity of gear 2, which is in mesh with gear 1. To do this, we use the formula for speed at internal gearing: v2 = v1 * r1 / r2, where v2 is the speed of the points on gear 2, r2 is the radius of the gear 2. Since gear 1 and the crank rotate independently of each other, the speed of point B will be equal to the speed of point C located on gear 2. Thus, v2 = v_C = 60 cm/s. Substituting the values into the formula for speed with internal gearing, we get:
w2 = w1 * r1 / r2 w2 = 2 * 30 / r2 w2 = 60 / r2
In order to find the angular velocity of gear 2, it is necessary to find the value of its radius r2. To do this, we use the formula for the length of the arc: l = r * alpha, where l is the length of the arc, alpha is the angle in radians. Since the length of the crank OA is 20 cm, the angle at which it rotates is alpha = 20 / 30 = 2 / 3 radians.
Using the formula for arc length, we find the length of the circular arc on which point C is located:
l = r2 * alpha l = 2/3 * r2
It is also known that the length of the arc on gear 2, on which point C is located, is equal to the length of the arc on gear 1, on which point B is located. That is:
l = r1 * delta, where delta is the engagement angle of the gear teeth.
Thus we get:
2/3 * r2 = 30 * delta
Let us express the angle of engagement of the gears through the radius of the wheel 2:
delta = (2/3 * r2) / 30
Let's substitute this expression into the formula for the angular velocity of gear 2:
w2 = 60 / r2 = 2 * 30 / [(2/3 * r2) / 30]
Let's simplify the expression:
w2 = 9 rad/s
Thus, the angular velocity of gear 2 is 9 rad/s.
This solution is a digital product that will help you understand problem 9.2.11 from the collection of Kepe O.?. in physics. The solution is made using formulas and principles of mechanics, and provides a detailed calculation of the angular velocity of gear 2 in an internal gear mechanism.
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This product is a solution to problem 9.2.11 from a collection of problems in differential mechanics, authored by O.?. Kepe. The problem describes a differential mechanism with internal gearing, consisting of gear 1, crank OA and gear 2. The problem is to determine the angular velocity of gear 2 with known angular velocities of gear 1 and crank OA.
To solve the problem, it is necessary to use the radius of the gear wheel 1, which is equal to 30 cm, and the length of the crank OA, which is equal to 20 cm. From the conditions of the problem, the angular speeds of rotation of the gear wheel 1 and the crank OA are known, which are equal to 2 rad/s and 4 rad/s, respectively. With. By solving the equations, you can get the answer to the problem, which is equal to 2 rad/s.
Thus, this product is a complete solution to problem 9.2.11 from the collection O.?. Kepe on differential mechanics, including all necessary calculations and explanations.
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