*Note: Type-setting limitations do not allow for arrows on the vector labels, so we have used boldface only.Concepts and Skills to Review
Net force: vector addition (Section 2.4 )
Gravitational forces (Section 2.5 )
Newton's second law: force and acceleration (Section 3.4 )
Motion with constant acceleration (Section 3.5 )
Falling objects (Section 3.6 )
Summary
Addition of vectors is commutative; A + B = B + A.
Vectors are added graphically by moving each vector so that its tail is placed at the tip of the previous vector. The sum is drawn as a vector arrow from the tail of the first vector to the tip of the last.
Vectors are subtracted graphically be drawing the vectors with their tails at the same point. Then the difference A - B is a vector drawn from the tip of B to the tip of A.
Addition and subtraction of vectors algebraically using components is generally easier and more accurate than the graphical method. The graphical method is still a useful first step to get an approximate answer.
To add vectors algebraically, add their components to find the components of the resultant: if A + B = C, then Ax + Bx = Cx and Ay + By = Cy.
Components are found by drawing a right triangle with the vector as the hypotenuse and the other two sides parallel to the x- and y-axes. Trigonometric functions are used to find the magnitudes of the components. The correct algebraic sign must be supplied for each component. The same triangle can be used to find the magnitude and direction of a vector if its components are known.
The x- and y- axes are chosen to make the problem easiest to solve. Any choice is valid as long as the two are perpendicular. In an equilibrium problem, choose x- and y-axes so that the fewest number of force vectors have to be resolved into both x- and y-components. In a non-equilibrium problem, if the direction of the acceleration is known, choose x- and y-axes so that the acceleration vector is parallel or antiparallel to one of the axes.
Theinstantaneous velocity vector is tangent to the path of motion.
Theinstantaneous acceleration vector does not have to be tangent to the path of motion since velocities can change both in direction and in magnitude.
The kinematic equations for an object moving in two dimensions with constant acceleration are:
Problems involving Newton's second law-whether equilibrium or nonequilibrium-can be solved by treating the x-and y-components of the forces and the acceleration separately. The vector equation Fnet = ma is equivalent to the two scalar equations
ΣFx= max and ΣFy= may
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For a projectile or any object moving with constant acceleration in the ±y-direction, the motion in the x- and y-directions can be treated separately. Since ax = 0, vxis constant. Thus the motion is a superposition of constant velocity motion in the x-direction on constant acceleration motion in the y-direction.
To relate the velocities of objects measured in different reference frames, use the equation
vAC = vAB + vBC
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where vAC represents the velocity of A relative to C, and so forth.
