{"id":141,"date":"2021-02-26T12:51:19","date_gmt":"2021-02-26T17:51:19","guid":{"rendered":"http:\/\/pressbooks.library.upei.ca\/statics\/?post_type=chapter&p=141"},"modified":"2025-07-25T13:28:56","modified_gmt":"2025-07-25T17:28:56","slug":"review","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.upei.ca\/statics\/chapter\/review\/","title":{"raw":"1.1 Preparatory Concepts","rendered":"1.1 Preparatory Concepts"},"content":{"raw":"

1.1.1 Scalar vs. Vector<\/h1>\r\n
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Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, \u201ca class period lasts 50 min\u201d or \u201cthe gas tank in my car holds 65 L\u201d or \u201cthe distance between two posts is 100 m.\u201d A physical quantity that can be specified completely in this manner is called a\u00a0<\/span>scalar quantity<\/strong>. Scalar is a synonym of \u201cnumber.\u201d Time, mass, distance, length, volume, temperature, and energy are examples of\u00a0<\/span>scalar<\/strong>\u00a0<\/span>quantities.<\/p>\r\n

Scalar quantities that have the same physical units can be added or subtracted according to the usual rules of algebra for numbers. For example, a class ending 10 min earlier than 50 min lasts (50 min - 10 min) = 40 min<\/span>. Similarly, a 60-cal serving of corn followed by a 200-cal serving of donuts gives (60 cal + 200 cal) = 260 cal<\/span>\u00a0<\/span>of energy. When we multiply a scalar quantity by a number, we obtain the same scalar quantity but with a larger (or smaller) value. For example, if yesterday\u2019s breakfast had 200 cal of energy and today\u2019s breakfast has four times as much energy as it had yesterday, then today\u2019s breakfast has 4(200 cal) = 800 cal<\/span>\u00a0<\/span>of energy. Two scalar quantities can also be multiplied or divided by each other to form a derived scalar quantity. For example, if a train covers a distance of 100 km in 1.0 h, its speed is 100.0 km\/1.0 h = 27.8 m\/s, where the speed is a derived scalar quantity obtained by dividing distance by time.<\/p>\r\n

Many physical quantities, however, cannot be described completely by just a single number of physical units. For example, when the U.S. Coast Guard dispatches a ship or a helicopter for a rescue mission, the rescue team must know not only the distance to the distress signal, but also the direction from which the signal is coming so they can get to its origin as quickly as possible. Physical quantities specified completely by giving a number of units (magnitude) and a direction are called\u00a0<\/span>vector quantities<\/strong>. Examples of vector quantities include displacement, velocity, position, force, and torque. In the language of mathematics, physical vector quantities are represented by mathematical objects called\u00a0<\/span>vectors<\/strong>. We can add or subtract two vectors, and we can multiply a vector by a scalar or by another vector, but we cannot divide by a vector. The operation of division by a vector is not defined.<\/p>\r\n\"A\r\n\r\nSource: University Physics Volume 1, OpenStax CNX https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/2-1-scalars-and-vectors<\/a>\r\n\r\n<\/div>\r\n

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Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nBasically<\/em>: a scalar has only magnitude, whereas a vector has magnitude and<\/span> direction.\r\n\r\nApplication:<\/em> I might have gone for a 4 km walk (scalar), but whether I walked in a straight line, took turns, or went 2 km out and turned around to walk 2 km back would tell me a lot more information (vector).\r\n\r\nLooking ahead: <\/em>We will talk about this again in sections 1.3 on vectors and in sections 1.4 and 1.5 on dot products and cross products.\r\n\r\n<\/div>\r\n<\/div>\r\n

1.1.2 Newton's Laws<\/h1>\r\n
\r\n\r\n1st Law:<\/strong>\r\n\r\nNewton's first law states that:\u00a0<\/span>\"A body at rest will remain at rest unless acted on by an unbalanced force. A body in motion continues in motion with the same speed and in the same direction unless acted upon by an unbalanced force.\"<\/strong>\r\n\r\nThis law, also sometimes called the \"law of inertia\", means that bodies maintain their current velocity unless a force is applied to change that velocity. If an object is at rest with zero velocity, it will remain at rest until some force begins to change that velocity, and if an object is moving at a set speed and in a set direction, it will remain at that same velocity until some force begins to change that velocity.\r\n\r\nNet Forces:\u00a0<\/strong>It is important to note that the\u00a0<\/span>net force<\/strong> is what will cause a change in velocity. The net force is the sum of all forces acting on the body. For example, we can imagine gently pushing on the rock in the figure below and observing that the rock does not move. This is because we will have a friction force equal in magnitude and opposite in direction opposing our gentle pushing force. The sum of these two forces will be equal to zero; therefore, the net force is zero, and the change in velocity is zero.<\/span>\r\n

Rotational Motion:\u00a0<\/strong>Newton's first law also applies to moments and rotational velocities. A body will maintain its current rotational velocity until a net moment is exerted to change that rotational velocity. This can be seen in things like toy tops, flywheels, stationary bikes, and other objects that will continue spinning once started until brakes or friction stop them.<\/span><\/p>\r\n\r\n\r\n[caption id=\"attachment_2653\" align=\"aligncenter\" width=\"300\"]\"An In the absence of friction in space, this space capsule will maintain its current velocity until some outside force causes that velocity to change. Public Domain image by NASA.[\/caption]\r\n\r\n \r\n\r\n[caption id=\"attachment_2654\" align=\"aligncenter\" width=\"300\"]\"A This rock is at rest with zero velocity and will remain at rest until a net force causes the rock to move. The net force on the rock is the sum of any force pushing the rock and the friction force of the ground on the rock opposing that force. Image by Liz Gray CC-BY-SA 2.0.[\/caption]\r\n\r\n \r\n\r\n[caption id=\"attachment_2655\" align=\"aligncenter\" width=\"300\"]\"A In the absence of friction, this spinning top would continue to spin forever, but the small frictional moment exerted at the point of contact between the top and the ground will slow the tops spinning over time. Image by Carrotmadman6 CC-BY-2.0[\/caption]\r\n\r\n \r\n\r\nSource: Engineering Mechanics, Jacob Moore et al., Mechanics Map - Newton's First Law<\/a>\r\n\r\n \r\n\r\n2nd Law:<\/strong>\r\n\r\nNewton's second law states that:\u00a0<\/span>\"When a net force acts on any body with mass, it produces an acceleration of that body. The net force will be equal to the mass of the body times the acceleration of the body.\"<\/strong>\r\n\r\n \r\n\r\nF<\/mi>\u2192<\/mo><\/mover><\/mrow>=<\/mo>m<\/mi>a<\/mi>\u2192<\/mo><\/mover><\/mrow><\/math> \r\n\r\nYou will notice that the force and the acceleration in the equation above have an arrow above them. This means that they are vector quantities, having both a magnitude and a direction. Mass, on the other hand, is a scalar quantity having only a magnitude. Based on the above equation, you can infer that the magnitude of the net force acting on the body will be equal to the mass of the body times the magnitude of the acceleration, and that the direction of the net force on the body will be equal to the direction of the acceleration of the body.\r\n\r\nRotational Motion:\u00a0<\/strong>Newton's second law also applies to moments and rotational velocities. The revised version of the second law equation states that the net moment acting on the object will be equal to the mass moment of inertia of the body about the axis of rotation (I) times the angular acceleration of the body.<\/span>\r\n\r\n \r\n\r\nM<\/mi>\u2192<\/mo><\/mover><\/mrow>=<\/mo>I<\/mi>\u2217<\/mo>\u03b1<\/mi>\u2192<\/mo><\/mover><\/mrow><\/math> \r\n\r\nYou should again notice that the moment and the angular acceleration of the body have arrows above them, indicating that they are vector quantities with both a magnitude and direction. The mass moment of inertia, on the other hand, is a scalar quantity having only a magnitude. The magnitude of the net moment will be equal to the mass moment of inertia times the magnitude of the angular acceleration, and the direction of the net moment will be equal to the direction of the angular acceleration.<\/span>\r\n\r\nSource: Engineering Mechanics, Jacob Moore et al., Mechanics Map - Newton's Second Law<\/a>\r\n\r\n \r\n\r\n3rd Law:<\/strong>\r\n\r\nNewton's Third Law states, <\/span>\"For any action, there is an equal and opposite reaction.\"<\/strong> By \"action,\" Newton meant a force, so for every force one body exerts on another body, that second body exerts a force of equal magnitude but opposite direction back on the first body. Since all forces are exerted by bodies (either directly or indirectly), all forces come in pairs, one acting on each of the bodies interacting.<\/span>\r\n\r\nThough there may be two equal and opposite forces acting on a single body, it is important to remember that for each of the forces, a Third Law pair acts on a separate body. This can sometimes be confusing when there are multiple Third Law pairs at work. Below are some examples of situations where multiple Third Law pairs occur.<\/span>\r\n\r\n[caption id=\"attachment_2656\" align=\"aligncenter\" width=\"260\"]\"A This volleyball resting on a surface has two pairs of Third Law forces. The first consists of the gravitational forces (one force on the ball and one force on the ground). The second consists of the normal forces at the point of contact (one force on the ball and one force on the ground). Image adapted from Public Domain image, no author listed.[\/caption]\r\n\r\n \r\n\r\n[caption id=\"attachment_2657\" align=\"aligncenter\" width=\"269\"]\"A If we ignore the weight of the two objects, this clamp will also have two pairs of Third Law forces. The first will be a set of normal forces at the top point of contact (one force on the wood and one force on the clamp) and the second will be another set of normal forces at the bottom point of contact (one force on the wood and one force on the clamp) Image adapted from Public Domain image, no author listed.[\/caption]\r\n\r\n \r\n\r\nSource: Engineering Mechanics, Jacob Moore et al., Mechanics Map - Newton's Third Law<\/a>\r\n\r\n<\/div>\r\n
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Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nBasically<\/em>: These 3 laws form the foundation of statics and dynamics. It makes our problems interesting! In statics, we don't do a lot with rotation.\r\n