## Kinetic Energy

Kinetic energy is a measure of the state of motion of an object. Kinetic energy is zero when the object is at rest. It is a scalar variable and is usually represented by the symbol T. For a particle of mass m, T is given by the following equation:

where v is the speed of the particle. Kinetic energy of a system of particles is given by the following equation:

where mi and vi are the mass and the speed of particle i, respectively, and the summation is over all particles in the collection.

Kinetic energy of a continuous body is given by the expression Figure 8.1. Velocity v of a point in a rigid body is equal to the velocity of the center of mass (vc) plus the vector product of the angular velocity (m) and the position vector connecting the center of mass to the point under consideration (r).

Figure 8.1. Velocity v of a point in a rigid body is equal to the velocity of the center of mass (vc) plus the vector product of the angular velocity (m) and the position vector connecting the center of mass to the point under consideration (r).

where v is the velocity of a small mass element dm. Integration is over the whole body.

We have seen previously that the velocity of a point P in a rigid body can be written as v = vc + a X r (8.4)

where v and vc represent the velocities of point P and the center of mass C, respectively, a is the angular velocity of the rigid body, and r is the position vector from C to P (Fig. 8.1).

In the case of planar motion of a rigid body, Eqns. 8.3 and 8.4 result in a simple expression for the kinetic energy:

in which Ic is the moment of inertia with respect to the center of mass. The kinetic energy has two identifiable parts: the kinetic energy of the center of mass and the kinetic energy of rotation relative to the center of mass.

As an illustration of this expression, let us consider the kinetic energy of a spherical ball rolling without a slip on a planar surface (Fig. 8.2). Let v e1 denote the velocity of the center of mass of the ball and let a be its radius. The no-slip requirement means that the velocity of that point of the sphere which touches the planar surface must be equal to zero:

Noting that for a sphere of radius a and mass m, Ic = (2/5) ma2, the kinetic energy of the spherical ball rolling without slip is equal to

T = (1/2) m (v)2 + (V2) (2/5) ma2 (—v/a)2 = 0.7 m v2

This equation shows that more than two-thirds of the kinetic energy is associated with the translational motion of the center of mass. Figure 8.2. Rolling of a spherical ball along an inclined plane. The symbol v denotes the velocity of the center of mass of the ball. The symbol m is as usual the angular velocity. Figure 8.2. Rolling of a spherical ball along an inclined plane. The symbol v denotes the velocity of the center of mass of the ball. The symbol m is as usual the angular velocity.

In some important problems of human body dynamics, the body will pivot around a point O. The kinetic energy of a rigid body that rotates around a fixed point O is given by the following equation:

where Io is the moment of inertia of the body with respect to an axis that passes through point O and is perpendicular to the plane of the motion. Let us illustrate the use of this equation by considering the kinetic energy of a pendulum. Let a slender rod of length L swing about a pivot O as shown in Fig. 8.3. We can compute the kinetic energy of the rod by using Eqn. 8.6:

Equation 8.5 can also be used to compute the same kinetic energy:

Figure 8.3. The rotation of a rod of length L with respect to point O, which is stationary in the reference frame E. The expression for kinetic energy of a body in three-dimensional motion contains more terms than that of planar motion. The definition of angular velocity in three-dimensional motion is presented in Chapter 9. However, it suffices to say that Eqn. 8.4 that relates the velocity of a point in a rigid body to the velocity of another point in the same body still holds. An expression for the kinetic energy of a rigid body in three dimensions can then be obtained by substituting Eqn. 8.4 into Eqn. 8.3:

T = (1/2> m (vc)2 + (1/2> [Icil «l2 + Ic22 «22 + Ic33 «32

+ 2Ici2 «1 0)2 + 2Ici3 «1 «3 + 2Ic23 «2 «3] (8.7)

in which «i is the angular velocity in the direction of i (i = 1, 2, 3), and the terms Iczy (i = 1, 2, 3, and j = 1, 2, 3) represents the components of the mass moment of inertia tensor. The mass moment of inertia is a set of geometric parameters; it depends only on the shape of the object and the distribution of mass within the object. Mathematically, the mass moment of inertia tensor is defined as

Ic11 = /(X22 + X32) pdV; Ic22 = /(X12 + X32) pdV; Ic33

Ic12 = -/(X1X2) pdV; Ic13 = -/(X1X3) pdV; Ic23 = -/(X2X3) pdV

in which p is the mass density (kg/m3), V is volume, and Xi is the distance along the axis ei, as measured from the center of mass of an object. For a slender rod of mass m and length L, the components of the moment of inertia with respect to the coordinate frame shown in Fig. 8.4 can be written as

Ic11 = 0, Ic22 = Ic33 = (mL2/12); Ic12 = Ic13 = Ic23 = 0

Components of the moment of inertia for geometrically simple homogeneous bodies are listed at the end of this volume.

Figure 8.4. A Cartesian coordinate system (x1, x2, X3) that is attached to the center of mass of a slender rod. 