A force acts about a catapult's point of rotation in order to hurl an object through the air, often as a weapon. The driving force of the catapult is best measured as a "moment," or the amount of rotational force imparted to the catapult's arm. The resultant force on the projectile is a function of the rotational and tangential accelerations that the arm induces on it. Note that both the moment and resulting force on the projectile will vary throughout the catapult's motion.
- Skill level:
- Moderately Challenging
Things you need
Calculate the moment on the catapult's arm. The moment is equal to the force acting perpendicular to the catapult's arm times its distance from the arm's point of rotation. If the force is supplied by a weight, the perpendicular force is equal to the weight times the sine of the angle between the weight's cable and the catapult arm. Sine is a trigonometric function.
Calculate the polar moment of inertia of the catapult's arm. Polar moment of inertia is a measure of an object's resistance to rotation. The polar moment of inertia of a generic object equals the integral of each of its infinitesimal units of mass times the square each unit mass' distance from the point of rotation. The integral is a calculus-based function. You may wish to approximate the catapult arm as a uniform rod, in which case the polar moment of inertia would become one-third the mass of the arm times the square of its length:
I = (m * L^2) / 3.
Calculate the angular acceleration. The angular acceleration is easily found by dividing the moment at any point in time by the polar moment of inertia:
a = M / I.
Calculate the normal and tangential accelerations on the projectile. The tangential acceleration describes how fast the object's linear velocity is increasing, and equals the angular acceleration times length of the arm. The normal acceleration, also referred to as centripetal acceleration, acts perpendicular to the object's instantaneous velocity and equals velocity squared divided by the length of the arm:
a = (v^2) / L.
You can approximate velocity at any point in time by multiplying the time that has elapsed by average angular acceleration and arm length:
v = a * t * L.
Use Newton's Second Law---force equals mass times acceleration---to convert the accelerations of the object into forces induced by the catapult. Multiply both the tangential and normal components of acceleration by the object's mass to obtain two forces.
Combine the two components of force into a single resultant force. Because the normal and tangential forces act perpendicular to each other, you can use the Pythagorean Theorem to find the magnitude of the resultant force:
a^2 + b^2 = c^2, where "a" and "b" are component forces and "c" is the resultant.
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