# Force Vector and Equilibrium

1.1 Purpose

The purpose of this experiment is to give you a qualitative and quantitative feel for vectors and forces in equilibrium.

1.2

Introduction

An object that is not accelerating falls into one of three categories: • The object is static and is subjected to a number of diﬀerent forces which cancel each other out. • The object is static and is not being subjected to any forces. (This is unlikely since all objects are subject to the force of gravity of other objects.) • The object is moving with constant velocity. In this case, the object may be subject to a number of forces which cancel out or no force at all. This case is not considered in this lab. The category of physics problems that involve forces in static equilibrium is called statics. Physicists and engineers are subjected to static problems quite frequently. A few examples of these principles in use are seen in the design of bridges and the terminal velocity of a person falling through the air. Mathematically, forces in equilibrium are just a special case of Newton’s Second Law of Motion, which states that the sum of all forces is equal to the mass of the object multiplied by the acceleration of the object. The special case of forces in equilibrium (static), occurs when the acceleration of the object is zero. When this situation arises, Newton’s Law becomes: ΣF = 0 (1.1)

This equation simply states that the sum of all of the force vectors acting on an object is equal to zero when the object is in equilibrium (static or not accelerating). We will use an apparatus called a ’force table’ to demonstrate static forces and equilibrium. 1

Note that equation 1.1 is a ’vector’ equation. A vector is a quantity that has both magnitude and direction. A quantity which does not have direction is called a scalar quantity. In order to distinguish vectors from scalars, vectors are identiﬁed with either bold face or an arrow on the top of the letter: V or V . A vector can be represented by an arrow with length proportional to the magnitude. The direction of an arrow gives the direction of a vector. There are two basic ways to add vectors: graphically and by components. To add vectors graphically, you place the tail of one on the head of the other. A vector from the tail of the ﬁrst to the head of the second vector is the resultant (sum). For two vectors A and B, the resultant C =A + B is shown below. A C B

The second method of adding vectors is to add their components to get the components of the resultant vector. Y A Y = |A| sin θ A θ AX AY X A X = |A| cos θ

Once you ﬁnd all the component you add the x components to get the resultant x component and likewise for the y components. The magnitude of the resultant can be found from the Pythagorean theorem: 2 2 R = Rx + Ry (1.2) where Rx is the x component of the resultant and Ry is the y component of the resultant. The direction of the resultant (sum) can be found from: θ = tan−1 ( Ry ) Rx (1.3)

Note that calculator acrtan functions may not determine the correct quadrant of the vector. This is easily determined by the signs of the x and y components in the ratio.

1.3

:

Equipment

Force table, masses, weight holders, pulleys, string. 2

Figure 1.1: The photograph shows the force table with the the pulleys, weight hangers and slotted weights.

1.4

Procedure

The force table apparatus is shown in ﬁgure 1.1. A screw under the pulley is used to secure the pulley to the edge of the force table. Make sure the string goes over the wheel of the pulley. • Two Forces in equilibrium: Place one pulley at zero degrees on the force table and add 100 grams (0.1 kg) to the mass hanger at the end of the string. The total mass will be approximately 150 grams (0.15 kg) since the mass hanger has a mass of 50 grams (0.05kg). Weigh the hanger and mass to determine the exact mass. The position of this pulley at 0o should not be moved for the rest of the experiment....

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