science-2018:PS.FM.2
science-2018:PS.FM.2 Forces
Learning goal: Treat force as a vector measured in newtons; identify gravitational, frictional, normal and tension forces acting on an object and draw its force diagram; add forces in one dimension to find the net force; calculate weight from F_g = mg and interpret g as gravitational field strength; relate friction to the normal force; and use the field model for forces that act at a distance.
Force is a vector. A force is a push or a pull with a size and a direction. The SI unit is the newton, N. One newton of net force gives a 1 kg object an acceleration of 1 m/s², so a newton is the same as a kilogram-meter per second squared, kg·m/s². For a sense of scale, the weight of a medium apple is roughly 1 N and the weight of a 60 kg person on Earth is about 588 N. Forces can be measured with a spring scale or a force probe. In this course all forces on an object lie along one line at a time, so each is positive or negative, and the net force is the one-dimensional vector sum: add all the signed values (a force in the negative direction enters the sum as a negative number) and keep the sign of the result.
Four contact-and-field forces you must recognize. Gravitational force, also called weight, is the pull of a planet on an object; it acts at a distance and always points toward the planet's center, which we call down. Friction is a force between surfaces that opposes motion, or attempted motion, of one surface across the other. Kinetic friction acts while one surface slips across the other; static friction acts where the surfaces grip without slipping, as when a push has not yet started something moving or a wheel rolls without skidding; drag is the friction of air or water on an object moving through it. These are treated conceptually in this course and quantified only when a force diagram gives their values. A normal force exists wherever two solid surfaces are pressed together by other forces, such as a book resting on a table, a ladder leaning on a wall or a bat striking a ball; it is always a push at right angles to the surfaces. A tension force is the pull of a taut rope, cord, cable or wire on the object attached to it; a slack rope exerts none.
Force diagrams. A force diagram represents the object as a box or dot and draws one arrow for every force acting on it, pointing in the force's direction, with lengths in proportion to the sizes. Only forces on the object go on its diagram: the pull the object exerts on something else belongs on that other object's diagram. Label each arrow with the type of force and, when known, its value. A chair resting on a floor has two forces: gravity down and the floor's normal force up, equal in size. A hanging mass on a string has gravity down and tension up. A skydiver has gravity down and drag up; early in the fall gravity is larger, and once drag has grown to equal gravity the two balance and the skydiver falls at a constant velocity.
Net force in one dimension. Suppose a crate is pushed to the right with 50 N while kinetic friction acts to the left with 20 N, and the crate's weight of 98 N is matched by an upward normal force of 98 N. Taking right and up as positive: horizontal net force = +50 N + (−20 N) = +30 N to the right; vertical net force = +98 N + (−98 N) = 0. The crate's motion changes in the horizontal direction only. In this course, when both horizontal and vertical forces act, one of the two directions always has zero net force.
Weight, mass and gravitational field strength. Mass, in kilograms, measures how much matter an object contains and does not change with location. Weight is the gravitational force on that mass and does change with location. The two are proportional: F_g = mg, where g is the gravitational field strength at the object's location. Near Earth's surface g = 9.8 N/kg, meaning each kilogram is pulled with 9.8 N. So a 4.0 kg bag weighs 4.0 kg × 9.8 N/kg = 39.2 N. Writing g as 9.8 m/s² is the same number in different units: 1 N/kg equals 1 m/s², and 9.8 m/s² is exactly the free-fall acceleration measured in the previous lesson, which is no coincidence. If you measure the weights of several objects with a spring scale and plot weight against mass, the points lie on a straight line through the origin whose slope is g at the place where you measured. Invented data of 0.5, 1.0, 1.5 and 2.0 kg giving 4.9, 9.8, 14.7 and 19.6 N have slope 9.8 N/kg, as in this lesson's diagram.
A worked model: weight on another world. A 60 kg astronaut weighs 60 kg × 9.8 N/kg = 588 N on Earth. On Mars, where the surface gravitational field strength is about 3.7 N/kg, the same 60 kg would weigh about 60 × 3.7 = 222 N. On the Moon, where the field strength is about 1.6 N/kg, the weight would be about 96 N. The astronaut's mass is 60 kg in all three places; a balance comparing masses would read the same, while a spring scale, which measures the gravitational force, would read differently. The field-strength values here are rounded; use the values a table gives you, and treat the answers as predictions.
The field model. Gravity, electric and magnetic forces act without contact. The field model explains this: an object with the right property (mass for gravity, charge for electric fields, a magnetic object for magnetic fields) has a field, a region of influence, around it. A second object placed in that field experiences a force. Two points from the model matter in this course. First, the stronger the field at a location, the greater the force on an object placed there; g = 9.8 N/kg is just the strength of Earth's gravitational field near its surface, and the weaker Martian field gives smaller weights. Second, the field of an object is present whether or not anything is in it: Earth's field surrounds Earth even where there is nothing to pull. Detailed calculation of gravitational, electric and elastic forces is reserved for Physics.
Friction and the normal force. Push a block across a table at a steady speed with a spring scale and the scale reads the kinetic friction, because at constant velocity the pull balances friction. Stack masses on the block and the reading rises. For the same pair of dry surfaces, the kinetic friction force grows, to a good approximation, in proportion to the normal force pressing the surfaces together, and the ratio depends on the two materials. Friction does not depend much on the area of contact. This is a model that works well for a block sliding steadily on a dry table, not a universal law of friction (wet, lubricated, very soft or sticky surfaces behave differently), and it is a relationship you can establish in the laboratory; the named constant for the ratio is a Physics topic.
Investigation: friction against normal force. Materials: a wooden block with a hook, a spring scale reading in newtons, a set of masses, a level table, a balance. Steps: measure the block's mass; on the level table the normal force equals the block's weight, mg; pull the block horizontally at a slow constant velocity and read the spring scale, taking the average of three trials; add 1.0 kg and repeat; add another 1.0 kg and repeat. Record normal force and friction force for each load and plot friction against normal force. Invented results: normal forces of 9.8, 19.6 and 29.4 N with friction of 2.9, 5.9 and 8.8 N give a nearly straight line through the origin with slope about 0.30. A straight line through the origin supports the claim that, for these two dry surfaces, kinetic friction is proportional to the normal force. What the investigation cannot show: the maximum static friction, which is usually larger than kinetic friction for the same surfaces and must be measured at the moment the block starts to move, and the effect of speed, which this method holds fixed.
Worked example 1 — the force diagram of a hanging mass. A 2.0 kg mass hangs at rest from a string. Forces on the mass: gravity, F_g = mg = 2.0 kg × 9.8 N/kg = 19.6 N downward, and the string's tension upward. Because the mass is at rest, the net force is zero, so the tension must be 19.6 N upward. The diagram shows one box with two equal arrows, down labeled "gravity 19.6 N" and up labeled "tension 19.6 N". The pull of the mass on the string does not appear on this diagram; it acts on the string.
Worked example 2 — weight from mass, and mass from weight. A textbook has a mass of 1.5 kg. Its weight on Earth is F_g = 1.5 kg × 9.8 N/kg = 14.7 N. Reverse the problem: a spring scale reads 490 N when a person stands on it. The person's mass is m = F_g ÷ g = 490 N ÷ 9.8 N/kg = 50 kg. Units check: N ÷ (N/kg) = kg. On Mars, with g ≈ 3.7 N/kg, the same person would weigh about 50 kg × 3.7 N/kg = 185 N and still have a mass of 50 kg.
Worked example 3 — net force from a diagram. A sled of weight 196 N (mass 20 kg) is pulled along level snow by a rope with tension 60 N to the right while kinetic friction of 20 N acts to the left; the snow pushes up with a normal force of 196 N. Take right and up as positive. Horizontal: +60 N + (−20 N) = +40 N, so the net force is 40 N to the right. Vertical: +196 N + (−196 N) = 0. The sled accelerates to the right; the size of that acceleration is the next lesson's topic. If the rope were slack, tension would be 0 and the only horizontal force would be −20 N of friction, slowing the sled.
Worked example 4 — a skydiver's diagram at two moments. Just after leaving the aircraft, a 70 kg skydiver has gravity of 70 × 9.8 = 686 N downward and only a small drag upward, say 100 N. Net force: −686 + 100 = −586 N, downward, so she speeds up. Later, drag has grown to 686 N. Net force: −686 + 686 = 0, so her velocity no longer changes and she falls at a constant velocity. Both diagrams have the same two arrows; only the length of the drag arrow differs, and that difference decides whether the motion changes.
Representation task: six force diagrams. Draw a labeled force diagram, with arrow lengths in proportion, for each of the following, and beneath it write the net force with its sign and the reason: (a) a 3.0 kg book resting on a table; (b) a 3.0 kg mass hanging at rest from a cord; (c) a 10 kg crate pushed right with 40 N against 15 N of kinetic friction on a level floor; (d) a skydiver at constant velocity; (e) a ladder leaning against a wall (qualitative: gravity, the floor's normal force and friction, the wall's normal force); (f) a sled being pulled up a slope by a rope, with friction (qualitative directions only). Use g = 9.8 N/kg for the numerical cases (a), (b) and (c); for (d) show only that the two forces are equal, and for (e) and (f) show directions only (in (e) take the wall as smooth, so it exerts no friction). Label each force by type: gravity, normal, friction, tension or drag.
Independent performance: weight, mass and field strength report. Using a spring scale and a balance, measure the mass and weight of at least five objects between 0.1 kg and 2.0 kg, or use a teacher-provided data set. Plot weight in newtons against mass in kilograms, draw the best straight line and find its slope. Success criteria: the table lists mass and weight with units for every object; the graph has labeled axes and a best-fit line through the origin; the slope is reported in N/kg and compared with 9.8 N/kg with a reason for any difference; the report explains what the slope represents (the gravitational field strength where the measurements were taken) and predicts, with working, the weight of one of the objects on a planet whose field strength is given as 3.7 N/kg; and it states in two sentences why a balance would read the same mass on both planets while a spring scale would not.
An adult may read the prompts aloud. Answer independently with words, objects, drawings or a demonstration before opening the answer guide. Award 1 point for a correct result or claim and 1 point for a matching explanation or demonstration; where a question asks only for a result (a corrected sentence, a spelling, a count), a fully correct result earns both points, and the answer guide shows what a complete response contains. Record prompting separately. The total (two points per question) is local practice feedback, not an Ohio proficiency level.
A 6.0 kg box rests on the floor. Draw (in words) its force diagram, give the value of each force, and state the net force.
Two forces act on the box: gravity, F_g = 6.0 kg × 9.8 N/kg = 58.8 N downward, and the floor's normal force, 58.8 N upward, a push at right angles to the floor. They are equal and opposite, so the net force is 0 N and the box stays at rest. The box's push down on the floor is not drawn because it acts on the floor, not the box.
Explain the difference between mass and weight, and calculate both for a 12 kg dog on Earth and on a moon where the gravitational field strength is 1.6 N/kg.
Mass measures the amount of matter and is 12 kg everywhere. Weight is the gravitational force on that mass, F_g = mg, and depends on the field strength. On Earth: 12 kg × 9.8 N/kg = 117.6 N, about 118 N. On the moon described: 12 kg × 1.6 N/kg = 19.2 N. The weight fell because the field is weaker there, not because the dog lost matter.
A rope pulls a 10 kg cart to the right with 35 N while kinetic friction of 15 N acts to the left on a level floor. Find the horizontal and vertical net forces.
Take right and up as positive. Horizontal: +35 N + (−15 N) = +20 N, so the net force is 20 N to the right. Vertical: gravity is 10 kg × 9.8 N/kg = 98 N down and the floor's normal force is 98 N up, so the vertical net force is 0. Only the horizontal motion changes.
What is a normal force, when does it exist, and in which direction does it act? Give two examples.
A normal force is the push between two solid surfaces that are pressed together by other forces. It exists whenever surfaces are in contact under load, such as a book on a table (table pushes up on the book) or a ladder leaning on a wall (wall pushes horizontally on the ladder). It always acts at right angles to the surfaces in contact, which is what "normal" means here.
Describe the field model for gravity and use it to explain why an object weighs less on Mars than on Earth even though its mass is unchanged.
In the field model, every object with mass is surrounded by a gravitational field, a region in which another mass experiences a force. The field is there whether or not anything is in it, and the stronger the field at a location, the greater the force on a mass placed there. Weight is F_g = mg, where g is the field strength. Mars's surface field is about 3.7 N/kg compared with Earth's 9.8 N/kg, so the same mass is pulled with a smaller force on Mars.
A student pulls a block at constant velocity across a table and the spring scale reads 3.0 N. She adds masses so the block's weight doubles and pulls it horizontally at constant velocity across the same table again. What reading is expected, and why?
About 6.0 N. At constant velocity the pull balances kinetic friction, so the scale reads the friction. On a level table the normal force equals the block's weight, and for the same two dry surfaces kinetic friction is approximately proportional to the normal force, so doubling the weight doubles the normal force and roughly doubles the friction. The area of contact does not matter much, only the force pressing the surfaces together.
A skydiver falls at constant velocity. Name the forces on her, compare their sizes and explain how the force diagram differs from the one just after she jumped.
At constant velocity the forces are gravity downward and drag upward, and they are equal in size, so the net force is zero. Just after jumping, drag was small because she was moving slowly, so gravity was larger than drag, the net force pointed down and she sped up. As speed rose, drag grew until it matched gravity. Both diagrams have the same two arrows; only the drag arrow's length changes.
Why does the pull of Earth on a book appear on the book's force diagram but the pull of the book on Earth does not?
A force diagram shows only the forces acting on the chosen object. Earth's pull acts on the book, so it belongs there. The book's pull acts on Earth, so it belongs on a diagram of Earth. Putting both on one diagram would wrongly suggest they cancel; they cannot, because they act on different objects.