Since point G is traveling in a horizontal circle at constant velocity we have centripetal acceleration. So we only need to consider the second and third equation. Since point G is traveling in a horizontal circle at constant velocity we have no tangential acceleration, so a GX = 0, and F GX = 0. M x is the moment acting in the local x-direction, at point G M y is the moment acting in the local y-direction, at point G M z is the moment acting in the local z-direction, at point G F GX is the force acting in the global X-direction, at point G F GY is the force acting in the global Y-direction, at point G F GZ is the force acting in the global Z-direction, at point GĪ GX is the acceleration of point G in the global X-directionĪ GY is the acceleration of point G in the global Y-directionĪ GZ is the acceleration of point G in the global Z-direction Note that a local xyz axes is defined as shown, and is attached to the wheel so that it moves with the wheel, and has origin at point G. A free-body diagram of the wheel (isolated from the rod) is given below. Let's analyze the forces and moments acting on the wheel, due to contact with the rod. To learn more about it visit the vector derivative page. Note that the terms dJ/dt and dK/dt (given above) are calculated using vector differentiation. The angular acceleration of the rod, with respect to ground, is zero since w r is constant and does not change direction. The angular velocity of the rod, with respect to ground, is The angular acceleration of the wheel, with respect to ground, is The angular velocity of the wheel, with respect to ground, is I, J, and K are defined as unit vectors pointing along the positive X, Y, and Z axis respectively. The global XYZ axes is fixed to ground and has origin at P. Where g is the acceleration due to gravity, point G is the center of mass of the wheel, and point P is the pivot location at the base. The general schematic for analyzing the physics is shown below. We will hence determine the equation of motion for the gyroscope. So now that we have an intuitive "feel" for the physics, we can analyze it in full using a mathematical approach. This can result in some interesting physics, such as a gyroscope not falling over due to gravity as it precesses. This may seem counter-intuitive, but the lesson here is that the acceleration of an object can act in a direction that is very different from the direction of motion. The acceleration of the particle is towards the center of the circle (centripetal acceleration), which is perpendicular to the velocity of the particle (tangent to the circle). This is the most basic explanation behind the gyroscope physics.Īs an analogy, consider a particle moving around in a circle at a constant velocity. In other words, due to the nature of the kinematics, the particles in the wheel experience acceleration in such a way that the force of gravity is able to maintain the angle θ of the gyroscope as it precesses. The force of gravity pulling down on the gyroscope creates the necessary clockwise torque M. Therefore a clockwise torque M is needed to sustain these forces. Due to Newton’s second law, this means that a net force F 1 must act on the particles in the top half of the wheel, and a net force F 2 must act on the particles in the bottom half of the wheel. The question is, why doesn't the gyroscope fall down due to gravity?!ĭue to the combined rotation w s and w p, the particles in the top half of the spinning wheel experience a component of acceleration a 1 normal to the wheel (with distribution as shown in the figure below), and the particles in the bottom half of the wheel experience a component of acceleration a 2 normal to the wheel in the opposite direction (with distribution as shown). Θ is the angle between the vertical and the rod (a constant)Īs the wheel spins at a rate w s, the gyroscope precesses at a rate w p about the pivot at the base (with θ constant). W p is the constant rate of precession, in radians/second W s is the constant rate of spin of the wheel, in radians/second To start off, let's illustrate a typical gyroscope using a schematic as shown below. Therefore, the physics of gyroscopes can be applied directly to a spinning top. Check out the video below of a toy gyroscope in action.Īs you've probably noticed, a gyroscope can behave very similar to a spinning top. But as it turns out, there is a fairly straightforward way of understanding the physics of gyroscopes without using a lot of math.īut before I get into the details of that, it's a good idea to see how a gyroscope works (if you haven't already). When people see a spinning gyroscope precessing about an axis, the question is inevitably asked why that happens, since it goes against intuition. Gyroscope physics is one of the most difficult concepts to understand in simple terms.
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