## Angular Acceleration

How do we determine the acceleration of a small volume element in a rigid object? We do this by taking the time derivative of the equation that relates the velocities of two points in a rigid object:

in which a is defined as the angular acceleration. In the planar motion, the rules concerning the sequential multiplication of three vectors lead to the following simplification for the third term in Eqn. 4.20b:

This equation indicates that the higher the rate of rotation of a limb, the larger the component of acceleration that is directed toward the center of rotation.

Example 4.3. Acceleration of the Center of Mass During Vertical Jumping. Let us apply Eqn. 4.20 to determine the acceleration of the center of mass of the athlete who is preparing for vertical jump. The details of this case were described in Example 4.1. The lengths of the various body segments of the athlete were given as Lf (length of the foot) = 27 cm, Li (length from ankle to knee) = 48 cm, Lt (length from knee to hip) = 50 cm, and Lc (length from hip to center of mass) = 28 cm. The athlete weighed 68

kg. The angles that various body segments made with the horizontal axis were presented as follows:

The angular velocity and angular acceleration of the various body segments can be deduced from these equations by taking the first and the second time derivative, respectively:

At time t = 0.2, the angles of rotatation, angular velocity, and angular acceleration are found to be

We next use these values in Eqn. 4.20b to determine the acceleration of the ankle, knee, hip, and finally the center of mass:

= -2 e3 X [-2 e3 X 0.27 (-cos 23° ei + sin 23° e2)] = 1 ei - 0.4 e2

aK = (1.0 e1 - 0.4 e2) + 40 e3 X 0.48 (cos 43° e1 + sin 43° e2) + 2 e3 X [2 e3 X 0.48 (cos 43° e1 + sin 43° e2)] = -13.4 e1 + 12.3 e2

aH = (-13.4 e1 + 12.3 e2) - 30 e3 X 0.50 (-cos 43° e1 + sin 43° e2)

-3 e3 X [-3 e3 X 0.50 (-cos 43° e1 + sin 43° e2)] = 0.1 e1 + 20.2 e2

X 0.28 (cos 52° e1 + sin 52° e2)] = -1.0 e1 + 18.8 e2

Let us next compute the ground force exerted on the athlete at time t = 0.2 s. According to the laws of motion, the mass of the athlete times the acceleration of the center of mass must be equal to the resultant force acting on the athlete: 