Force Force, Work, Energy, and Momentum Machinery's Handbook, 31st Edition
186
Accelerations Resulting from Unbalanced Forces.— In the section describing the reso lution and composition of forces, page 163 , it was stated that when the resultant of a system of forces is zero, the system is in equilibrium, that is, the body on which the force system acts remains at rest or continues to move with uniform velocity. If, however, the resultant of a system of forces is not zero, the body on which the forces act will be acceler- ated in the direction of the unbalanced force. To determine the relation between the un- balanced force and the resulting acceleration, Newton’s laws of motion must be applied. These laws may be stated as follows: First Law: Every body continues in a state of rest or in uniform motion in a straight line until it is compelled by a force to change its state of rest or motion. Second Law: Change of motion is proportional to the force applied and takes place along the straight line in which the force acts. The “force applied” represents the resultant of all the forces acting on the body. This law is sometimes worded: An unbalanced force acting on a body causes an acceleration of the body in the direction of the force and of magnitude proportional to the force and inversely proportional to the mass of the body. Stated as a formula, R = Ma where R is the resultant of all the forces acting on the body, M is the mass of the body (mass = weight W divided by acceleration due to gravity g ), and a is the acceleration of the body resulting from application of force R . Third Law: To every action there is always an equal reaction; or, if a force acts to change the state of motion of a body, the body offers a resistance equal and directly opposite to the force. Newton’s second law may be used to calculate linear and angular accelerations of a body produced by unbalanced forces and torques acting on the body; however, it is necessary first to use the methods described under Algebraic Composition and Resolution of Force Systems starting on page 163 to determine the magnitude and direction of the resultant of all forces acting on the body. Then, for a body moving with pure translation, R Ma g W a = = where R is the resultant force in pounds acting on a body weighing W pounds; g is the gravitational constant, usually taken as 32.16 ft/sec 2 , approximately; and a is the resulting acceleration in ft/sec 2 of the body due to R and in the same direction as R . Using metric SI units, the formula is R = Ma , where R = force in newtons (N), M = mass in kilograms (kg), and a = acceleration in meters/second squared (m/s 2 ). It should be noted that the weight of a body of mass M kg is Mg newtons, where g is approximately 9.81 m/s 2 . Free Body Diagram: In order to correctly determine the effect of forces on the motion of a body it is necessary to construct a free body diagram . This diagram shows 1) the body removed or isolated from contact with all other bodies that exert force on it; and 2) all the forces acting on the body. For example, in Fig. 1a the block being pulled up the plane is acted upon by certain forces; the free body diagram of this block is shown at Fig. 1b. Note that all forces acting on the block are indicated. These forces include: 1) the force of gravity (weight); 2) the pull of the cable, P ; 3) the normal component, W cos φ , of the force exerted on the block by the plane; and 4) the friction force, μ W cos φ , of the plane on the block.
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