Download Pushes and Pulls Content 3. Daily examples of force

Content 1. What are forces? 2. Measurement of a force 3. Daily life examples of forces Pushes and Pulls 4. Useful mathematics: Vectors 5. Newton’s laws of motion for IJSO training course 6. Free body diagram 7. Mass, weight and gravity 8. Density vs. mass 9. Turning effect of a force 1 2 1. What are forces? 2. Measurement of a force • Force, simply put, is a push or pull that an object exerts on another. • Force is measured in units called Newton (N). We can measure a force using a • We cannot see the force itself but we can observe what it can do: spring balance (彈簧秤). – It can produce a change in the motion of a body. The body may change in speed or direction. – It can change the shape of an object. A force is the cause of velocity change or deformation. The SI unit of force: N (Newton) 3 • Many materials including springs extend evenly when stretched by forces, provided that the force is not too large. This is known as Hooke’s law (虎
克定律). (Wikimedia commons) 4 3. Daily examples of force Weight • A spring balance uses the extension of a spring to measure force. The extension is proportional to the force acting on it as shown below. • The weight of an object is the gravitational force acting on it. Weight
5 6 Normal force Tension • A book put on a table does not fall because its weight is balanced by another force, the normal force, from the table. • Tension (張力) in a stretched string tends to shorten it back to the original length. • Once the string breaks or loosens, the tension disappears immediately. • Normal: perpendicular to the table surface. normal force • Since tension acts inward to shorten the string, we usually draw two “face­to­face” arrows to represent it. normal force force by the hand Draw “face­to­face” arrows to represent tension weight 7 8 Example Friction • Friction (摩擦力) arises whenever an object slides or tends to slide over another object. tension 10 N “face to face” arrows representing tension • It always acts in a direction opposite to the motion. These two forces counterbalance each other (suppose the weight of the hook is negligible). • Cause: No surface is perfectly smooth. When two surfaces are in contact, the tiny bumps catch each other. tension 10 N 1­kg mass The tension balances the weight, therefore the mass does not fall down. motion weight 10 N friction 9 Friction can be useful Friction drags motion. 10 • The tread patterns on tyres also prevents the car from slipping on slippery roads. Moreover, road surfaces are rough so as to prevents slipping of tyres. • We are not able to walk on a road without friction, which pushes us forward. • In rock­climbing, people need to wear shoes with studs. The studs can be firmly pressed against rock to increase the friction so that the climber will not slide easily.
Tread pattern on a car tyre (Wikimedia commons) 11 backward push of foot on road forward push of road on foot Tread pattern on a mountain bicycle tyre (Wikimedia commons)
12 Disadvantages of friction – using lubricating oil • There are some disadvantages of friction. For example, in the movable parts of machines, energy is wasted as sound and heat to overcome friction. Friction will also cause the wear in gears. – streamlining – using air cushion The streamlined shape cuts down the air­friction on the racing car. • Friction can be reduced by the following ways. – bearings BHC SR­N4 The world's largest car and passenger carrying hovercraft (Wikimedia commons) 1. Propellers 2. Air 3. Fan 4. Flexible skirt 13 14 (All pictures are from Wikimedia commons) 4. Useful mathematics: Vectors Example: displacement • A mouse moves 4 cm northward and then 3 cm eastward. • A scalar (標量) is a quantity that can be completely described by a magnitude (size). – Examples: distance, speed, mass, time, volume, temperature, charge, density, energy. • What is the distance travelled? – It is not sensible to talk about the direction of a scalar: the temperature is 30 o C to the east(?). • What is the displacement of the mouse? 3 cm – Answer = 4 cm + 3 cm = 7 cm • A vector (向量) is a quantity that needs both magnitude and direction to describe it. 4 cm 5 cm – Answer = 5 cm towards N36.9 o E – Examples: displacement, velocity, acceleration, force. How to find the angle? A vector has a direction. 15 16 Example: velocity • A bird is flying 4 m/s northward. There suddenly appears a wind of 3 m/s blowing towards the east. Example: force • You push a cart with 4 N towards north. Your friend helps but he pushes it with 3 N towards the east. 3 m/s • What is the velocity of the bird? – Answer = 5 m/s towards N36.9 o E 3 N • What is the resultant force? 4 m/s – Answer = 5 N towards N36.9 o E 5 m/s • What is the speed of the bird? 4 N 5 N • What is the magnitude of the force? – Answer = 5 m/s – Answer = 5 N • Note 1: No need to specify the direction. • Note: A magnitude does not have a direction. • Note 2: the answer is not simply = 3 m/s + 4 m/s = 7 m/s 17 A magnitude does not have a direction.
18 Addition and resolution • Two vectors can add up to form a single vector, a vector can also be resolved into two vectors. • Two usual ways to denote a vector • In physics, we usually resolved a vector into two perpendicular components. F F • Below, a force F is resolved into two components, F x and F y. – Boldface c – Adding an arrow • Vectors can be added by using the tip­to­tail or the parallelogram method. • If vectors a and b add up to become c, we can write c = a + b. b a Fx = F cos θ
Tip­to­tail method F y = F sin θ
b c tan θ =
a Parallelogram method F y F x F = F x + F y 2 19 2 20 5. Newton’s laws of motion • Isaac Newton developed three laws of motion, which give accurate description on the motion of cars, aircraft, planet, etc. • Newton’s 3 laws of motion answer 3 questions: – If the cue does not hit the ball, what will happen to the ball? • The laws are important but simple. They are just the answers to three simple questions. • Newton’s first law – If the cue hits the ball, what will happen to the ball? • Consider a cue and a ball. • Newton’s second law – If the cue hits the ball, what will happen to the cue? • Newton’s third law 21 22 The second law The first law • The acceleration of an object is directly proportional • Also called “The law of inertia” (慣性定律) to, and in the same direction as, the unbalanced force acting on it, and inversely proportional to the mass of the object. • A body continues in a state of rest or uniform motion in a straight line unless acted upon by some net force. • Galileo discovered this. • In the form of equation, the second law can be written as F = ma • If the cue does not hit the ball, the ball will remain at rest. – F is the acting force – m is the mass of the object – a is the acceleration (a vector) of the object • If the cue hits the ball, the ball will accelerate. 23 Second law: F = ma
24 But .. what is acceleration? • Consider an object moving from A to B in 2 hours with a uniform velocity. What is the velocity? N AB = 2 + 2 km = 2828 . 43 m 2
(Note: This AB does not have an arrow. It indicates a length, which is a scalar.) B (1 km, 3 km) N B (1 km, 3 km) Speed = AB / 7200 s = 0.39 m/s A (3 km, 1 km) Final displacement from O = OB 2 O (Note: speed is also a scalar.) A (3 km, 1 km) O E Initial displacement from O = OA E Velocity = 0.39 m/s towards NW. Change in displacement = OB – OA = AB Velocity = Change in displacement = Time required AB 2 hours Velocity = Change in displacement Time required 25 • Consider a bird. At time t = 0 s, N it was moving 5 m/s towards SE. Its velocity gradually changed such that at t = 2 s, its velocity became 5 m/s towards NE. 26 vc = N 5 2 + 5 2 m/s = 7 . 07 m/s (Note: This vc does not have an arrow. It indicates a magnitude.) v2 v2 vc = v2 ­ v1 vc = v2 ­ v1 Magnitdue of acceleration = vc / 2 s = 3.54 m/s 2 E E • Calculate the acceleration. v1 v1 Acceleration = 3.54 m/s 2 towards N. Change in velocity = vc Acceleration = Change in velocity Time required = vc 2 s Acceleration = Change in velocity Time required 27 28 Equations of motion in 1D Uniform acceleration • In the 1D, there are only two directions, left and right, up and down, back and forth, etc. • Let • t = the time for which the body accelerates • a = acceleration (which is assumed constant) • For these simple cases, once we have chosen a positive direction, we can use + and ­ signs to indicate direction. We can also use a symbol without boldface to denote a vector. • u = the velocity at time t = 0, the initial velocity • v = the velocity after time t, the final velocity • s = the displacement travelled in time t • We can prove that – Example: If we choose downward positive, the velocity v = ­5 m/s describes an upward motion of speed 5 m/s. 29 v = u + at 1 s = ut + at 2
2 v 2 = u 2 + 2 as 30 Back … to the second law: F = ma Displacement­time graph Velocity­time graph v • Mass is a measure of the inertia, the tendency of an object to maintain its state of motion. The SI unit of mass is kg (kilogram). s • 1 Newton (N) is defined as the net force that gives an acceleration of 1 m/s 2 to a mass of 1 kg. parabola slope = a u • The same formula can be applied to the weight of a body of mass m such that W = mg. – W: the weight of the body. It is a force, in units of N. 0 0 t t – g: gravitational acceleration = 9.8 m/s 2 downward, irrespective of m . W = mg 31 32 The third law Force of man accelerates the cart. • For every action, there is an equal and opposite reaction. • When the cue hits the ball, the ball also “hits” the cue. Action: the man pushes on the wall. Reaction: the wall pushes on the man. The same force accelerates two carts half as much. Twice as much force produces acceleration twice as much. Action: Earth pulls on the falling man. Reaction: The man pulls on Earth. 33 Example • The block does not fall because its weight is balanced by a normal force from the table surface. 34 Explanation Normal force = mg (upward) • Action and reaction act on different bodies. They cannot cancel each other. • The “partner” of the weight is the gravitational attraction of the block on the Earth. Weight = mg (downward) • Are the weight and the normal force an action­ and­reaction pair of force as described by Newton’s third law? Weight = mg (downward) Gravitational attraction of the block on the Earth = mg (upward)
• Answer: No! 35 36 Explanation 6. Free body diagram • The “partner” of the normal force acting on the block by the table surface is the force acting on the table by the block surface. Normal force = mg (upward) • To study the motion of a single object in a system of several bodies, one must isolate the object and draw a simple diagram to indicate all the external forces acting on it. This diagram is called a free body diagram. • Both have the same magnitude mg. • But they do not cancel each other because they are acting on different bodies. Example The force acting on the table by the block = mg (downward) 37 N W Worked Example 1 For an object of mass m at rest on a table surface, there are two external forces acting on it: 1. Its weight W 2. Normal force from the table surface N. Obviously, W = ­N, and W = N = mg. 38 Solution: Method 1 Take rightward positive. Let a be the acceleration of the blocks. Let f be the pushing force on Block B by Block A. • Consider two blocks, A and B, on a smooth surface. • Find – (a) the pushing force on Block B by Block A. Consider the free body diagram of Block A – (b) the acceleration of the blocks. normal force from the table surface 10 N F Block A Block B 3 kg 2 kg 3 kg 10 N a f (reaction force of the pushing force on Block B) weight 39 a 10 N 3 kg 40 Then consider the free body diagram of Block B normal force from the table surface f a f Vertical direction: No motion. The weight and the normal force from the table balance each other. Horizontal direction: Applying Newton’s second law (F = ma), we have (with units neglected) 10 ­ f = 3a 2 kg weight We ignore the vertical direction because the forces are balanced. Consider the horizontal direction. Applying the second law again, we have (1) f = 2a 41 (2)
42 Solution: Method 2 We now have 2 equations in 2 unknowns. 10 ­ f = 3a (1) • Method 1 is a long method, below is a shorter one. f = 2a (2) • The whole system is a mass of 5 kg. • We take rightward positive and define the same f and a as those in Method 1. Solving them, we have f = 4 N • Applying the second law (F = ma), we have 10 = 5a, hence a = 2 m/s 2 . a = 2 m/s 2 • Consider only Block B. The only force acting on it is f. Hence f = 2a = 4 N. (a) The pushing force on Block B by Block A = 4 N towards the right. (b) The acceleration of the blocks = 2 m/s 2 . 43 44 Worked Example 2 Solution • Consider a pulley and two balls, A and B. For convenience, take g = 10 m/s 2 . Take downward positive. Let tension = T and acceleration of Ball A = a. • Find Consider the free body diagram of Ball A: T – (a) the acceleration of Ball A. – (b) the tension in the string. A: 4 kg a Weight = 4g We can apply F = ma and get A: 4 kg 4g ­ T = 4a B: 1 kg (1) 45 46 Worked Example 3
Consider the free body diagram of Ball B: T • Consider a block on an inclined plane. B: 1 kg • Label all forces acting on the block and resolve them into components parallel and perpendicular to the plane. a Weight = g We apply F = ma and get g ­ T = ­a (2) Solving Equations (1) and (2), we get a = 6 m/s 2 and T = 16 N. (a) The acceleration of Ball A = 6 m/s 2 downward. (b) The tension in the string = 16 N. 47 48 • Find the acceleration a of the block in terms of g, given that 1
.
θ = 30 o , f = µR , µ =
2 3 Now, consider the motion parallel to the motion. Applying Newton’s second law F = ma, we have Solution Consider the motion perpendicular to the motion. The forces are balanced, therefore we have mg sin θ − f = ma mg sin 30 o − µR = ma R = mg cos θ
1 1 3 mg −
mg = ma 2 2 3 2 1 a = g 4 = mg cos 30 o
=
3 mg 2 49 50 7. Mass, weight and gravity • The mass of an object is a measure of its inertia. It is always the same wherever the object is. • In everyday life, people often confuse mass with weight. • On the other hand, the weight W of an object is the pull of the gravity acting on it. It depends on its mass m and the gravitational acceleration g. • A piece of meat does not weigh 500 g, but its mass is 500 g and it weighs about 5 N on the Earth. • W = mg • g varies slightly with positions on the Earth. • g is different on different celestial objects: (Wikimedia commons) Earth Moon Venus Jupiter 9.80665 m/s 2 1.622 m/s 2 8.87 m/s 2 24.79 m/s 2 51 52 • Only two forces are acting on the girl, Weightlessness – Weight of the girl = W • When a girl stands inside a lift, she cannot feel her own scale reading = R weight. What she feels is the normal force R acting on her by the lift floor. weight (W ) reaction (R) • The scale reading shows the magnitude of the reaction force to R, that is, the force acting on the scale by her feet. – Normal force acting on her = R • The scale reading (= R) is the girl’s apparent weight. weight (W ) reaction (R)
Apparent weight = R 53 • The motion of the lift can change R, and hence the girl will feel a different weight. • If the lift falls freely, R = 0, the girl will feel weightless. She is in a state of weightlessness. 54 Measurement of density 8. Density vs. mass • To find the density of an object, one must know both the mass and volume. • Density (密度) is a commonly­used concept in daily life. We say, for example, a plastic foam board is less dense than a piece of metal. • Mass: can be measured by a balance. • Volume: How to measure? • Intuition tells us that more mass packed into a small volume will give a higher density. • Answer: • In fact, the density of an object is defined as Density = From the rise in level, we can measure the volume. mass volume 55 Measuring the density of an irregular solid Measuring the density of a liquid 56 9. Turning effect of a force • When we turn on a tap or open a door, the tap or the pivot door handle will rotate about an axis or a fixed point called the pivot.(支點). The perpendicular distance between the force and the pivot is called the moment arm (力臂). axis • The moment of a force is a measure of this turning (Wikimedia commons) effect. Moment is a vector quantity and its direction is indicated by either clockwise or anticlockwise. Its definition is Moment = Force × moment arm = Fd pivot 57 (Wikimedia commons) • Principle of moments (力矩原理) – When a body is in balance, the total clockwise moment about any point is equal to the total anticlockwise moment about the same point.
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