FORCES Mass, Force and Acceleration

FORCES

Mass, Force and Acceleration

Mass: The amount of matter that a body contains with the SI unit being the kilogram (kg)

Force: A force may be defined as a push or pull on a body. The SI unit for force is the Newton (N). Forces cannot be seen but their effects can be felt or seen. For example, when you go round a corner in a car, you do not see the forces but your body moves and you can feel the force acting on you.

Acceleration: is the rate of change of velocity, or in other words how long it takes something to speed up or slow down. Because acceleration is a fundamental element in Newton's second law it is pivotal in understanding the effect of gravity and how mass and force relate to each other. If a car moves from 0-100 km/h in 12 seconds and another car performs the same feat in 6 seconds the second car is accelerating twice as quickly. Since acceleration is a vector there can be negative acceleration: this is called deceleration. The SI unit for acceleration is metres per second squared (ms-2 or m/s2)

Gravity: The Earth exerts a gravitational force on all bodies. The amount of the

gravitational force depends on the mass of the objects and the distance between

them. For all objects on the earth's surface, the gravitational force accelerates at 9.8ms-2 towards the centre of the earth. In all calculations in this course acceleration due to gravity (g) is 9.8 ms-2, however some publications may use 10 ms-2.

Weight: Do not confuse mass with weight. Weight is a force that is created by gravity acting upon a mass. When you "weigh" yourself on the scales at home your weight force acts on the scales causing the scales to give a reading. But because we are so used to using kilograms as a measure of our weight, the scales are calibrated in kilograms instead of Newtons. This convention is misleading as weight is a force and the units must be Newtons. If you are on the moon your mass will be the same as it was on earth but your weight will be less, as the moon has less gravitational force than the earth. In space you have no weight as there is no gravity but your mass is the same as it was on earth.

Weight is the effect that the earth's gravitational force has on a body. The weight of a body is the force created by the product of a body's mass and the acceleration due to gravity. Thus the formula for weight is a variation of Newton's second law. Therefore,

W mg

W = weight (N)

m = mass (kg) g = acceleration due to gravity (ms-2)

This means that any mass will have a unit in kilograms, while a weight will have a unit in Newtons. If you want to know what your true weight is, multiply the reading you get off the scales (your mass) by 9.8. So if you have a mass of 60 kg, then your weight will be 588 N.

Example 1

A car has a mass of 1100 kg, determine its weight.

W mg W 1100 9.8 W 10,780 N W 10.78 kN

W = ?

m = 1100 kg g = 9.8 ms-2

Example 2

A car has a mass of 1.5 tonnes. Four people are sitting in the car and their average mass is 78 kg. Determine the total weight of the car and its occupants.

Analysis: You need to convert the car's mass in tonnes to kilograms then add all the masses together. You cannot add dissimilar units.

W mg W (mC 4mO )g W (1500 (4 78)) 9.8 W 17,757.6 N W 17.76 kN

W = ? mC = mass of car = 1500 kg mO = mass of each occupant = 78 kg g = 9.8 ms-2

Review Questions

1. Determine the weight of a 68 kg cable on a bridge.

2. A truck has a mass of 18 tonnes. If a 400 kg object is placed on the back of the truck what is the total weight.

3. A bus has a mass of 8 tonnes. It carries 40 passengers with an average mass of 80 kg; determine the total weight of the bus.

4. A person weighs 750 N on Earth and is chosen to go to the Moon. If the acceleration due to gravity on the moon is 1.6 ms-2, calculate the weight of the astronaut on the Moon.

5. A truck goes onto a weight bridge, the two rear axles register a weight of 15 kN each and the front axle registers a weight on 10 kN, what is the total mass of the truck?

Scalar and Vector Quantities

Scalars are quantities that are fully described by a magnitude (or numerical value) alone.

Vectors are quantities that are fully described by both a magnitude and a direction.

Quantity

a. 5 m b. 30 m/sec, East c. 5 mi., North d. 20 degrees Celsius e. 256 bytes f. 4000 kJ

Circle the correct category

Scalar or Vector Scalar or Vector Scalar or Vector Scalar or Vector Scalar or Vector Scalar or Vector

Vector quantities are often represented by vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. Vector diagrams are used to depict the forces acting upon an object. Such diagrams are commonly called as free-body diagrams. An example of a scaled vector diagram is shown in the diagram at the right. The vector diagram depicts a displacement vector. Observe that there are several characteristics of this diagram that make it an appropriately drawn vector diagram.

a scale is clearly listed a vector arrow (with arrowhead) is drawn in a specified direction. The vector

arrow has a head and a tail. the magnitude and direction of the vector is clearly labelled. In this case, the

diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).

Vector Addition

Two vectors can be added together to determine the result (or resultant).

These rules for summing vectors were applied to free-body diagrams in order to determine the net force (i.e., the vector sum of all the individual forces).:

Addition of vectors can be extended to more complicated cases in which the vectors are directed in directions other than purely vertical and horizontal directions. For example, a vector directed up and to the right will be added to a vector directed up and to the left. The vector sum will be determined for the more complicated cases shown in the diagrams below.

There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The two methods are: Analytical: using the Pythagorean theorem and trigonometric methods Graphical: the head-to-tail method using a scaled vector diagram

The Pythagorean Theorem

To see how the method works, consider the following problem: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement. This problem asks to determine the result of adding two displacement vectors that are at right angles to each other. The result (or resultant) of walking 11 km north and 11 km east is a vector directed northeast as shown in the diagram to the right. Since the northward displacement and the eastward displacement are at right angles to each other, the Pythagorean theorem can be used to determine the resultant (i.e., the hypotenuse of the right triangle).

The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of 15.6 km.

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