Forces in One Dimension

Forces in One Dimension
Chapter Forces in One Dimension 4

Force and Motion 4.1 Force and Motion
Section Force and Motion 4.1 Force and Motion A force is defined as a push or pull exerted on an object. Forces can cause objects to speed up, slow down, or change direction as they move.

Section Force and Motion 4.1 Force and Motion Consider a textbook resting on a table. How can you cause it to move? Two possibilities are that you can push on it or you can pull on it. If you push harder on an object, you have a greater effect on its motion. The direction in which force is exerted also matters. If you push the book to the right, the book moves towards right. The symbol F is a vector and represents the size and direction of a force.

Section Force and Motion 4.1 Force and Motion When considering how force affects motion, it is important to identify the object of interest. This object is called the system. Everything around the object that exerts forces on it is called the external world. A contact force exists when an object from the external world touches a system and thereby exerts a force on it.

Force and Motion 4.1 Contact Forces and Field Forces
Section Force and Motion 4.1 Contact Forces and Field Forces If you drop a book, the gravitational force of Earth causes the book to accelerate. This is an example of a field force. Use the Force Obi-wan! They are typically invisible fields which is why their discover is relatively recent (last few hundred years) What are some others? Forces result from interactions; thus, each force has a specific and identifiable cause called the agent. Without both an agent and a system, a force does not exist. A model which represents the ALL forces acting ON a system, is called a free-body diagram or FBD for short.

Force and Motion 4.1 Force and Acceleration – A Proof for NSL
Section Force and Motion 4.1 Force and Acceleration – A Proof for NSL A stretched rubber band exerts a pulling force; the farther you stretch it, the greater the force with which it pulls back. Stretch the rubber band for a constant distance of 1 cm to exert a constant force on the cart. Now let it go.

Force and Motion 4.1 Force and Acceleration
Section Force and Motion 4.1 Force and Acceleration You can construct a graph as shown here. The graph indicates that the constant increase in the velocity is a result of the constant.

Section Force and Motion 4.1 Force and Acceleration Increase the force applied on the cart gradually. Plot a velocity-time graph for each 2 cm, 3 cm and so on and calculate the acceleration (slope). The relationship between the force and acceleration is linear, where the greater the force, the greater the resulting acceleration.

Section Force and Motion 4.1 Force and Acceleration To determine the meaning of the slope on the force- acceleration graph, increase the number of carts gradually. The acceleration of two carts is 1/2 the acceleration of one cart, and the acceleration of three carts is 1/3 the acceleration of one cart.

Section Force and Motion 4.1 Force and Acceleration The slope, k, is defined as the reciprocal of the mass The equation indicates that a force applied to an object causes the object to accelerate.

Section Force and Motion 4.1 Force and Acceleration The formula, , tells you that if you double the force, you will double the same object’s acceleration. If you apply the same force to several different objects, the one with the most mass will have the smallest acceleration and the one with the least mass will have the greatest acceleration. One unit of force causes a 1-kg mass to accelerate at 1 m/s2, so one force unit has 1 kg·m/s2 or one newton and is represented by N. Vector sum of all the forces on an object is net force.

Section Force and Motion 4.1 Newton’s Second Law Newton’s second law

Force and Motion 4.1 Newton’s Second Law
Section Force and Motion 4.1 Newton’s Second Law A useful strategy for finding how the motion of an object: First, identify all the forces acting on the object. Draw a free-body diagram showing the direction and relative strength of each force acting on the system. Then, add the forces to find the net force. Next, use Newton’s second law to calculate the acceleration. Finally, if necessary, use kinematics to find the velocity or position of the object.

Force and Motion 4.1 Newton’s First Law
Section Force and Motion 4.1 Newton’s First Law In the absence of a net force, the motion (or lack of motion) of both the moving object and the stationary object continues as it was. Newton recognized this and generalized Galileo’s results in a single statement. Newton’s First Law: “an object that is at rest will remain at rest, and an object that is moving will continue to move in a straight line with constant speed, if and only if the net force acting on that object is zero.” If the net force on an object is zero, then the object is in equilibrium.

Force and Motion 4.1 Newton’s First Law
Section Force and Motion 4.1 Newton’s First Law Newton’s first law is sometimes called the law of inertia – the lazy force Inertia is the tendency of an object to resist change. If an object is at rest, it tends to remain at rest. If it is moving at a constant velocity, it tends to continue moving at that velocity.

Section Force and Motion 4.1 Newton’s First Law

Section Check 4.1 Question 1
Two horses are pulling a 100-kg cart in the same direction, applying a force of 50 N each. What is the acceleration of the cart? 2 m/s2 1 m/s2 0.5 m/s2 0 m/s2

Section Check 4.1 Answer 1 Answer: B
Reason: If we consider positive direction to be the direction of pull then, according to Newton’s second law,

Two friends Mary and Maria are trying to pull a 10-kg chair in opposite directions. If Maria applied a force of 60 N and Mary applied a force of 40 N, in which direction will the chair move and with what acceleration? The chair will move towards Mary with an acceleration of 2 m/s2. The chair will move towards Mary with an acceleration of 10 m/s2. The chair will move towards Maria with an acceleration of 2 m/s2. The chair will move towards Maria with an acceleration of 10 m/s2.

Section Check 4.1 Answer 2 Answer: C
Reason: Since the force is applied in opposite direction, if we consider Maria’s direction of pull to be positive direction then, net force = 60 N – 40 N = 20 N . Thus, the chair will move towards Maria with an acceleration.

Section Check 4.1 Question 3 State Newton’s first law.
What is it in ONE word?

Section Section Check 4.1 Answer 3 Newton’s first law states that “an object that is at rest will remain at rest, and an object that is moving will continue to move in a straight line with constant speed, if and only if the net force acting on that object is zero”. Inertia

Using Newton's Laws 4.2 Using Newton’s Second Law
Section Using Newton's Laws 4.2 Using Newton’s Second Law Newton’s second law tells you that the weight force, Fg, exerted on an object of mass m is Why is Fg, or weight, location specific? Hint: you weight less on the moon, right? Consider a free-falling ball in midair. It is touching nothing and air resistance can be neglected, the only force acting on it is Fg. Both the force and the acceleration are downward – speeds up! Force causes acceleration which is Δvelocity! BAM!!

Using Newton's Laws 4.2 Using Newton’s Second Law
Section Using Newton's Laws 4.2 Using Newton’s Second Law How does a bathroom scale work? When you stand on the scale, the spring in the scale exerts an upward force on you because you are in contact with it. Because you are not accelerating, the net force acting on you must be zero (Fnet=ma=0). The spring force, Fsp, upwards must be the same magnitude as your weight, Fg, downwards.

Using Newton's Laws 4.2 Fighting Over a Toy
Section Using Newton's Laws 4.2 Fighting Over a Toy Anudja is holding a stuffed dog, with a mass of 0.30 kg, when Sarah decides that she wants it and tries to pull it away from Anudja. If Sarah pulls horizontally on the dog with a force of 10.0 N and Anudja pulls with a horizontal force of 11.0 N, what is the horizontal acceleration of the dog?

Section Using Newton's Laws 4.2 Fighting Over a Toy Sketch the situation (FBD) and identify the dog as the system and the direction in which Anudja pulls as positive.

Section Using Newton's Laws 4.2 Fighting Over a Toy Identify known and unknown variables. Known: m = 0.30 kg FAnudja on dog = 11.0 N FSarah on dog = 10.0 N Unknown: a = ?

Section Using Newton's Laws 4.2 Fighting Over a Toy Substitute FAnudja on dog = 11.0 N, FSarah on dog = 10.0 N, m = 0.30 kg

Section Using Newton's Laws 4.2 Apparent Weight

Using Newton's Laws 4.2 Drag Force and Terminal Velocity
Section Using Newton's Laws 4.2 Drag Force and Terminal Velocity When an object moves through any fluid, such as air or water, the fluid exerts a drag force on the moving object in the direction opposite to its motion. As the ball’s velocity increases, so does the drag force. The constant velocity that is reached when the drag force equals the force of gravity is called the terminal velocity.

If mass of a person on Earth is 20 kg, what will be his mass on moon? (Gravity on Moon is six times less than the gravity on Earth.)

Reason: Mass of an object does not change with the change in gravity, only the weight changes.

Your mass is 100 kg, and you are standing on a bathroom scale in an elevator. What is the scale reading when the elevator is falling freely?

Reason: Since the elevator is falling freely with acceleration g, the contact force between elevator and you is zero. As scale reading displays the contact force, it would be zero.

In which of the following cases will your apparent weight be greater than your real weight? The elevator is at rest. The elevator is accelerating in upward direction. The elevator is accelerating in downward direction. Apparent weight is never greater than real weight.

Reason: When the elevator is moving upwards, your apparent weight (where m is your mass and a is the acceleration of the elevator). So your apparent becomes more than your real weight.

Interaction Forces 4.3 Identifying Interaction Forces
Section Interaction Forces 4.3 Identifying Interaction Forces When you exert a force on your friend to push him forward, he exerts an equal and opposite force on you, which causes you to move backwards. An interaction pair of forces is two forces that are in opposite directions and have equal magnitude.

Interaction Forces 4.3 Identifying Interaction Forces
Section Interaction Forces 4.3 Identifying Interaction Forces An interaction pair is also called an action-reaction pair of forces. This might suggest that one causes the other; however, this is not true. The two forces either exist together or not at all.

Interaction Forces 4.3 Newton’s Third Law
Section Interaction Forces 4.3 Newton’s Third Law Newton’s third law, which states that all forces come in pairs. Newton’s Third Law states that the force of A on B is equal in magnitude and opposite in direction of the force of B on A. The two forces in a pair act on different objects and are equal in magnitude and opposite in direction. They are vectors!

Interaction Forces 4.3 Earth’s Acceleration
Section Interaction Forces 4.3 Earth’s Acceleration When a softball with a mass of 0.18 kg is dropped, its acceleration toward Earth is equal to g, the acceleration due to gravity. What is the force on Earth due to the ball, and what is Earth’s resulting acceleration? Earth’s mass is 6.0×1024 kg.

Section Interaction Forces 4.3 Earth’s Acceleration Draw free-body diagrams for the two systems: the ball and Earth and connect the interaction pair by a dashed line.

Section Interaction Forces 4.3 Earth’s Acceleration Identify known and unknown variables. Known: mball = 0.18 kg mEarth = 6.0×1024 kg g = 9.80 m/s2 Unknown: FEarth on ball = ? aEarth = ?

Section Interaction Forces 4.3 Earth’s Acceleration Use Newton’s second and third laws to find aEarth.

Section Interaction Forces 4.3 Earth’s Acceleration Substitute a = –g

Section Interaction Forces 4.3 Earth’s Acceleration Substitute mball = 0.18 kg, g = 9.80 m/s2

Section Interaction Forces 4.3 Earth’s Acceleration Use Newton’s second and third laws to solve for FEarth on ball and aEarth.

Section Interaction Forces 4.3 Earth’s Acceleration Substitute FEarth on ball = –1.8 N

Section Interaction Forces 4.3 Earth’s Acceleration Use Newton’s second and third laws to find aEarth.

Section Interaction Forces 4.3 Earth’s Acceleration Substitute Fnet = 1.8 N, mEarth= 6.0×1024 kg

Interaction Forces 4.3 Forces of Ropes and Strings
Section Interaction Forces 4.3 Forces of Ropes and Strings The force exerted by a string or rope is called tension. At any point in a rope, the tension forces are pulling equally in both directions. You cannot PUSH a rope.

Interaction Forces 4.3 The Normal Force
Section Interaction Forces 4.3 The Normal Force The normal force is a perpendicular contact force exerted by a surface on another object. It exists because the object has mass and accelerates towards the Earth. This is the force that keeps you in your chair or a book from falling to the ground. What happens when the normal forces is smaller than the objects weight? The normal force is critical when calculating friction forces.

Explain Newton’s third law – statement, equation, and what happens when the applied force is larger then the reaction force that can be generated?

Section Section Check 4.3 Answer 1 Suppose you push your friend, the force of you on your friend is equal in magnitude and opposite in direction to the force of your friend on you. This is summarized in Newton’s third law, which states that forces come in pair. The two forces in a pair act on different objects and are equal in strength and opposite in direction. Newton’s third law The force of A on B is equal in magnitude and opposite in direction of the force of B on A. Something fails. It will break.

If a stone is hung from a rope with no mass, at which place on the rope will there be more tension? The top of the rope, near the hook. The bottom of the rope, near the stone. The middle of the rope. The tension will be same throughout the rope.

Section Check 4.3 Answer 2 Answer: D
Reason: Because the rope is assumed to be without mass, the tension everywhere in the rope is equal to the stone’s weight .

In a tug-of-war event, both teams A and B exert an equal tension of 200 N on the rope. What is the tension in the rope? In which direction will the rope move? Explain with the help of Newton’s third law.

Section Section Check 4.3 Answer 3 Team A exerts a tension of 200 N on the rope. Thus, FA on rope = 200 N. Similarly, FB on rope = 200 N. But the two tensions are an interaction pair, so they are equal and opposite. Thus, the tension in the rope equals the force with which each team pulls (i.e. 200 N). According to Newton’s third law, FA on rope = FB on rope. The net force is zero, so the rope will stay at rest as long as the net force is zero.

Friction 5.2 Static and Kinetic Friction
Section Friction 5.2 Static and Kinetic Friction There are two types of friction, and both always oppose motion. Kinetic friction is exerted when the two surfaces rub against each other. To understand the other friction, imagine trying to push a heavy couch across the floor but it does not move. Newton’s 3rd law tell you that there must be a second horizontal force acting on the couch, one that opposes your force and is equal in size. This force is static friction, which is the force exerted on one surface by another when there is no motion between the two.

Section Friction 5.2 Static and Kinetic Friction You might push harder and harder but if the couch still does not move, the force of friction must be getting larger. This is because the static friction force acts in response to other forces.

Section Friction 5.2 Static and Kinetic Friction When you push hard enough the couch will begin to move. There is a limit to how large the static friction force can be. Once your force is greater than this maximum static friction, the couch begins moving and kinetic friction begins to act on it instead.

Section Friction 5.2 Static and Kinetic Friction Frictional force depends on the materials that the surfaces are made of. There is more friction between skis and concrete than there is between skis and snow. The normal force between the two objects also matters. The harder one object is pulled (weight) against the other, the greater the force of friction.

Section Friction 5.2 Static and Kinetic Friction If you pull a block along a surface at a constant velocity, according to Newton’s laws, the frictional force must be equal and opposite to the force with which you pull. You can pull a block of known mass at a constant velocity and use a spring scale to measure the force that you pull. You can then stack additional blocks on the block to increase the normal force and repeat.

Section Friction 5.2 Static and Kinetic Friction There is a direct proportion between the kinetic friction force and the normal force. The different lines correspond to dragging the block along different surfaces. Note that the sandpaper surface has a steeper slope than the polished table.

Section Friction 5.2 Static and Kinetic Friction The slope of this line, designated μk, is called the coefficient of kinetic friction between the two surfaces and relates the frictional force to the normal force. Kinetic friction force

Friction 5.2 Static and Kinetic Friction Static Friction Force
Section Friction 5.2 Static and Kinetic Friction Static Friction Force In the equation for the maximum static friction force, μs is the coefficient of static friction between the two surfaces, and μsFN is the maximum static friction force that must be overcome before motion can begin.

Section Friction 5.2 Static and Kinetic Friction Note that the equations for the kinetic and maximum static friction forces involve only the magnitudes of the forces. The forces themselves, Ff and FN, are at right angles to each other.

Friction 5.2 Balanced Friction Forces
Section Friction 5.2 Balanced Friction Forces You push a 25.0 kg wooden box across a wooden floor at a constant speed of 1.0 m/s. How much force do you exert on the box?

Section Friction 5.2 Balanced Friction Forces Identify the forces and establish a coordinate system.

Section Friction 5.2 Balanced Friction Forces Draw a motion diagram indicating constant v and a = 0.

Friction 5.2 Balanced Friction Forces Draw the free-body diagram.
Section Friction 5.2 Balanced Friction Forces Draw the free-body diagram.

Section Friction 5.2 Balanced Friction Forces Identify the known and unknown variables. Known: m = 25.0 kg v = 1.0 m/s a = 0.0 m/s2 μk = 0.20 Unknown: Fp = ?

Section Friction 5.2 Balanced Friction Forces The normal force is in the y-direction, and there is no acceleration. FN = Fg = mg

Section Friction 5.2 Balanced Friction Forces Substitute m = 25.0 kg, g = 9.80 m/s2 FN = 25.0 kg(9.80 m/s2) = 245 N

Section Friction 5.2 Balanced Friction Forces The pushing force is in the x-direction; v is constant, thus there is no acceleration. Fp = μkmg

Section Friction 5.2 Balanced Friction Forces Substitute μk = 0.20, m = 25.0 kg, g = 9.80 m/s2 Fp = (0.20)(25.0 kg)(9.80 m/s2) = 49 N

Section Check 5.2 Question 1 Define friction force. Section
Did you think to ask which one first?

Section Section Check 5.2 Answer 1 A force that opposes motion is called friction force. There are two types of friction force: Kinetic friction—exerted on one surface by another when the surfaces rub against each other because one or both of them are moving. Static friction—exerted on one surface by another when there is no motion between the two surfaces.

Juan tried to push a huge refrigerator from one corner of his home to another, but was unable to move it at all. When Jason accompanied him, they where able to move it a few centimeter before the refrigerator came to rest. Which force was opposing the motion of the refrigerator? Static friction Kinetic friction Before the refrigerator moved, static friction opposed the motion. After the motion, kinetic friction opposed the motion. Before the refrigerator moved, kinetic friction opposed the motion. After the motion, static friction opposed the motion.

Reason: Before the refrigerator started moving, the static friction, which acts when there is no motion between the two surfaces, was opposing the motion. But static friction has a limit. Once the force is greater than this maximum static friction, the refrigerator begins moving. Then, kinetic friction, the force acting between the surfaces in relative motion, begins to act instead of static friction.

Section Check 5.2 Question 3 On what does a friction force depends?
The material that the surface are made of The surface area Speed of the motion The direction of the motion

Section Check 5.2 Answer 3 Answer: A
Reason: The materials that the surfaces are made of play a role. For example, there is more friction between skis and concrete than there is between skis and snow.

Section Check 5.2 Question 4 – 4!!! How could I do that to you!
A player drags three blocks in a drag race, a 50-kg block, a 100-kg block, and a 120-kg block with the same velocity. Which of the following statement is true about the kinetic friction force acting in each case? Kinetic friction force is greater while dragging 50-kg block. Kinetic friction force is greater while dragging 100-kg block. Kinetic friction force is greater while dragging 120-kg block. Kinetic friction force is same in all the three cases. All of the above…

Reason: Kinetic friction force is directly proportional to the normal force, and as the mass increases the normal force also increases. Hence, the kinetic friction force will hit its limit while dragging the maximum weight.

Force and Motion in Two Dimensions
Section Force and Motion in Two Dimensions 5.3 Equilibrium Revisited Recall that when the net force on an object is zero, the object is in equilibrium. An object in equilibrium is motionless or moves with constant velocity. Equilibrium can occur no matter how many forces act on an object. As long as the resultant is zero, the net force is zero.

Section Force and Motion in Two Dimensions 5.3 Equilibrium Revisited What is the net force acting on the object? Vectors may be moved if you do not change their direction (angle) or length.

Section Force and Motion in Two Dimensions 5.3 Equilibrium Revisited The addition of the three forces, A, B, and C. There is no net force; thus, the sum is zero and the object is in equilibrium.

Section Force and Motion in Two Dimensions 5.3 Equilibrium Revisited Suppose that two forces are added are not zero. How could you find a third force that would result in zero, and therefore cause the object to be in equilibrium? Find the sum of the two forces already on the object. This single force that produces the same effect as the two individual forces added together, is called the resultant force. The force that you need to find is one with the same magnitude as the resultant force, but in the opposite direction. A force that puts an object in equilibrium is called an equilibrant.

Section Force and Motion in Two Dimensions 5.3 Equilibrium Revisited Finding the equilibrant for two vectors. This works for any number of vectors.

Section Force and Motion in Two Dimensions 5.3 Motion Along an Inclined Plane

Section Force and Motion in Two Dimensions 5.3 Motion Along an Inclined Plane Because an object’s acceleration is usually parallel to the slope, one axis, usually the x-axis, should be in that direction – the ramp. With this coordinate system, there are two forces—normal and frictional forces. These forces are in the direction of the coordinate axes. However, the weight is not –> vector components. SOH, CAH, TOA. When an object is placed on an inclined plane, the normal force will not be equal to the object’s weight. You will need to apply Newton’s laws once in the x-direction and once in the y-direction.

If three forces A, B, and C are exerted on an object as shown in the following figure, what is the net force acting on the object? Is the object in equilibrium?

Section Section Check 5.3 Answer 1 We know that vectors can be moved if we do not change their direction and length. The three vectors A, B, and C can be moved (rearranged) to form a closed triangle. Since the three vectors form a closed triangle, there is no net force. Thus, the sum is zero and the object is in equilibrium. An object is in equilibrium when all the forces add up to zero.

How do you decide the coordinate system when the motion is along a slope? Is the normal force between the object and the plane the object’s weight?

Section Section Check 5.3 Answer 2 An object’s acceleration is usually parallel to the slope. One axis, usually the x-axis, should be in that direction. The y-axis is perpendicular to the x-axis and perpendicular to the surface of the slope. With these coordinate systems, you have two forces—the normal force and the frictional force. Both are in the direction of the coordinate axes. However, the weight is not. This means that when an object is placed on an inclined plane, the magnitude of the normal force between the object and the plane will usually not be equal to the object’s weight.

A skier is coming down the hill. What are the forces acting parallel to the slope of the hill? Normal force and weight of the skier. Frictional force and component of weight of the skier along the slope. Normal force and frictional force. Frictional force and weight of the skier.

Reason: The component of the weight of the skier will be along the slope, which is also the direction of the skier’s motion. The frictional force will be in the opposite direction from the direction of motion of the skier.

Forces in Two Dimensions
Chapter Forces in Two Dimensions 5 End of Chapter 4 & 5

Forces in One Dimension

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