Torque, Angular momentum and Equilibrium

Transcription

1 Torque, Angular momentum and Equilibrium 1. Torque: the moment of force Consider a particle M of position vector r moving in a force field F, then we can define the torque of the force F about the origin O as: τ = r F (1) Which is a vector quantity of magnitude r F sin θ, where θ is the angle between the two vectors r and F, or θ = r, F. Note that τ is perpendicular to both r and F. Figure 1. The unit of torque is N m. Although its dimension is the same as that of the work, but they are quite different quantities. The torque is a vector quantity while work is a scalar. Note that the torque τ depends not only on the force F, but also on the choice of the reference O. 2. Center of gravity The center of gravity of a system of particles is a very important concept in mechanics. Sometimes a body s weight is treated like single force 1

2 acting at a point of the body known as the center of gravity. The center of gravity may be determined by the equilibrium equation or by experiment. If gravity is uniform (which is the case if we study the gravitational field in a small area on our planet), the center of mass is the same as center of gravity. For a discrete system of N particles of masses m 1, m 2, m 3,..., m N, with corresponding position vectors r 1, r 2, r 3,..., r N, the position of the center of mass (also the center of gravity in case of uniform gravity) is given by the position vector r C.M. : r C.M. = r C.G. = m 1 r 1 + m 2 r m N r N m 1 + m m N = N i=1 m i r i N i=1 m i For continuous systems, the discrete sum becomes intergral: r rdm rdm C.M. = = dm M = N i=1 m i r i M (2) (3) The integral in this equation can be over a volume (for a system of volume V), or over a surface (for a layer of surface S), or a line (for a linear system of length l). 3. Angular momentum: The angular momentum L of a point mass m, moving with velocity v, is by definition the moment of its linear momentum m v with respect to a point O: L = r p (4) where r is the position vector of the particle m, and p = m v is its linear momentum. L is a vector quantity, and depends on the choice of the origin O. Figure 2. The unit of angular momentum in SI system is kg m/s or J s. 2

3 3.1. The relation between angular momentum and torque The derivative of the angular momentum with respect to time gives: Remember that and Then dt = d d r ( r p) = p + r dt dt dt d r dt = v dt = F dt = v (m v) + r F But v and m v are parallel vectors. Then their vector product is zero. So dt = r F (5) But r F is the torque τ of the force applied on the particle. Then τ = d L dt (6) Note the analogy between this equation and Newton s second law: F = dt 4. Conservation of angular momentum If the total external torque acting on a particle is zero, that is τ = 0, then: dt = 0 L = const. (7) Hence, if the total external torque acting on a particle is zero, then its angular momentum is conserved. Note the analogy between this statement and the statement of conservation of linear momentum, which states: if the total external forces acting on a particle is zero, then its linear momentum is conserved. That is: dt = F ext = 0 p = const. 3

4 5. Angular momentum in central forces 5.1. Central force A central force is a force F having two main characteristics: 1. It is directed towards or away from a fixed point O, which could often be chosen to be the origin. 2. Its magnitude depends only on the distance r from the fixed point O. So if we choose the fixed point - from which the central force F is originated - to be the origin of coordinates, then we can summarize the above two characteristics in the following expression of F: F = f (r) ur (8) where f (r) a function of r, its absolute value f (r) is the magnitude of the force F, and ur is the unit vector defined in the direction of the position vector r. Note that if f (r) < 0, then the force F is directed towards O; if f (r) > 0, then F is directed away from O Angular momentum in central field Let us study the angular momentum of a particle in the presence of a central force field F. Consider a particle of mass m moving in a central force field F. Let us choose the origin of the force field lines to be the origin of coordinates O. Then the torque τ on the particle due to the force F is τ = r F Since the force F is central, it is parallel to the position vector r (since ur is parallel to r). Then their vector product is zero. So Hence τ = r F = r [ f (r) ur ] = f (r) r u r = 0 (9) dt = τ = 0 L = const. (10) So L is a constant vector. Hence, a particle subjected to a central force, has constant angular momentum, when the angular momentum is measured relative to the origin of the force field lines. 6. Equilibrium of rigid bodies A rigid body, is a body in which the distance between any two particles does not change during motion; otherwise the system is called elastic. The forces acting on a rigid body do not change its shape and size. However 4

5 they may change its state from one at rest, for example, to one of translational or rotational velocity. The body is said to be in equilibrium if the external forces acting on it do not produce any change in its translational or rotational state Translational equilibrium The body is said to be in translational equilibrium if it remains at rest or moves with a uniform velocity along a straight line. This leads to the first condition of equilibrium: When the body is in translational equilibrium, it is said to satisfy the first condition of equilibrium. In this case, the sum of all the external forces acting on the body is zero: F ext = 0 (11) 6.2. Rotational equilibrium The body is said to be in rotational equilibrium if it does not rotate or rotates at constant angular velocity, which leads to the second condition of equilibrium: When the body is in rotational equilibrium, it is said to satisfy the second condition of equilibrium. In this case, the sum of all the external torques acting on the body is zero: τ ext = 0 (12) 6.3. Equilibrium of a rigid body A body is said to be in equilibrium, if it satisfies the first and the second conditions of equilibrium. That is: { F ext = 0 (13) τ ext = 0 Physics Zone by Farid Minawi 5