Forces & Momentum

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レッスンノート

Free-Body Diagrams

A free-body diagram shows all the forces acting on a single object, represented as a point particle at its centre of mass.
Each force is drawn as a vector arrow: the length represents magnitude and the direction shows the force's direction.
Forces are labelled with conventional symbols such as Fg (weight), FN (normal contact force), FT (tension) and Ff (friction).
Common forces to include: weight always acts towards the planet's surface; tension pulls away from the mass; normal contact force acts perpendicular to a surface; friction opposes motion.
The resultant force is the vector sum of all forces acting on the body and determines the change in its motion.
Forces can be balanced (resultant force zero) or unbalanced (non-zero resultant force).
In one dimension, combine forces by adding them with sign according to direction; in two dimensions, use Pythagoras or trigonometry to find the resultant.

Forces on an object

Newton's First Law

Newton's first law: a body remains at rest or moves with constant velocity unless acted on by a resultant force.
If the resultant force on an object is zero, it is in translational equilibrium and must be either at rest or moving at a constant velocity.
A resultant force changes an object's motion by speeding it up, slowing it down, or changing its direction.
For balanced forces, the forces to the left equal the forces to the right, and upward forces equal downward forces.
Constant velocity means zero acceleration, so the forces must be balanced even if the object is moving.

Examples of Newton's First Law

Newton's Second Law

Newton's second law: the resultant force on an object is directly proportional to its acceleration, written as F = ma.
The acceleration always acts in the same direction as the resultant force.
If the resultant force acts along the direction of motion, the object speeds up; if it opposes motion, the object slows down.
If the resultant force acts at an angle to the direction of motion, the object changes direction.
Newton's second law can also be expressed as: the resultant force equals the rate of change of momentum, F = Δp/Δt.
For a falling object with no drag, acceleration is independent of mass.

Newton's second law in action

Newton's Third Law

Newton's third law: if Object A exerts a force on Object B, then Object B exerts a force on Object A that is equal in magnitude and opposite in direction.
Forces in a third-law pair must be the same type, have the same magnitude, act in opposite directions, and act on different objects.
Example: a foot pushes the ground backwards, and the ground pushes the foot forwards; both are normal contact forces.
A free-body diagram showing weight and normal contact force on the same object is an example of Newton's first law, not third law.
To apply Newton's third law, consider the interaction between two objects: e.g., the book pushes on the table and the table pushes back on the book.

Contact and Non-Contact Forces

A contact force acts between objects that are physically touching; examples include friction, fluid resistance (viscous drag), tension, and normal reaction force.
Surface friction opposes motion when surfaces rub against each other, e.g., car wheels on the ground.
Viscous drag (fluid resistance) occurs when an object moves through a liquid or gas; air resistance is a type of viscous drag.
Tension occurs within an object when pulling forces are applied to both ends.
The normal force is the component of the contact force perpendicular to the surface that counteracts the body.
A non-contact force acts at a distance due to a field; examples include gravitational, electrostatic, and magnetic forces.
Gravitational force is the attractive force between objects with mass; weight is Fg = mg.
Electrostatic and magnetic forces can be attractive or repulsive.

Frictional Forces

Frictional forces oppose motion: they can prevent a stationary object from moving or reduce the speed of a moving object.
Friction transfers energy by heating, raising the thermal energy of the objects and surroundings.
Static friction occurs when a body is stationary on a surface; it matches any push or pull until it can no longer hold the object stationary.
Dynamic friction occurs when a body is in motion on a surface and has a constant value for a given situation.
For any given situation, the maximum static friction is larger than dynamic friction.
The equation for static friction is Ff ≤ μs FN; for dynamic friction it is Ff = μd FN.
The coefficient of friction is a number between 0 and 1 (not including 0 or 1) and depends on the two surfaces.

Hooke's Law

Hooke's law: the extension of a material is directly proportional to the applied force up to the limit of proportionality.
Hooke's law is expressed as FH = −kx, where k is the spring constant (N m⁻¹) and x is the extension (m).
The spring constant measures the stiffness of a material; a larger k means a stiffer material.
Extension is the difference between the stretched length and the unstretched length: extension = stretched length − unstretched length.
A force–extension graph for a material obeying Hooke's law is a straight line through the origin; the gradient of this linear region equals the spring constant k.
If the axes are swapped (force on x-axis, extension on y-axis), the gradient is 1/k instead.

Stoke's Law

Viscous drag is the frictional force between an object and a fluid that opposes their relative motion.
Stoke's law gives the viscous drag force on a sphere: Fd = 6πηrv.
In the equation, η is the fluid viscosity (N s m⁻² or Pa s), r is the sphere's radius (m), and v is its velocity (m s⁻¹).
The drag force depends on the speed, size, and shape of the object, and on the viscosity of the fluid.
Viscosity is a measure of a fluid's resistance to flow; fluids with high viscosity are difficult to pour.
The rate of flow of a fluid is inversely proportional to its coefficient of viscosity.

Conservation of Linear Momentum

Momentum is the product of mass and velocity: p = mv; it is a vector quantity.
The fundamental SI units of momentum are kg m s⁻¹ (or N s).
The conservation of linear momentum states that total momentum remains constant in a closed system with no external forces.
For collisions or explosions in one dimension, total momentum before = total momentum after.
In an elastic collision, both momentum and kinetic energy are conserved.
In an inelastic collision, momentum is conserved but kinetic energy is not conserved (some is transferred to other forms).

Conservation of momentum

Impulse and Force

Impulse is the product of force and the time for which it acts: Impulse = FΔt.
Impulse is also equal to the change in momentum: FΔt = Δp.
The area under a force–time graph gives the impulse.
Newton's second law in terms of momentum: the resultant force equals the rate of change of momentum, F = Δp/Δt.
A crumple zone increases the impact time during a collision, which reduces the force exerted on the car and its passengers.

Comparison of holding an umbrella in rain versus hail, illustrating impulse.

Circular Motion

Objects moving in a circle have a constantly changing direction, so their velocity is constantly changing and they are accelerating.
Angular velocity ω is the rate of change of angle, measured in rad s⁻¹.
Centripetal acceleration is directed towards the centre of the circle and has magnitude a = v²/r or a = ω²r.
Centripetal force is the resultant force directed towards the centre that keeps an object moving in a circle, given by F = mv²/r or F = mω²r.
In non-uniform circular motion, the speed changes as well as the direction, so there is both a centripetal and a tangential component of acceleration.

Circular motion of the International Space Station

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練習問題

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1.Which of the following is the correct definition of a contact force?
Easy
- AA force which acts between objects that are physically touching
- BA force which acts at a distance without physical contact
- CA force that only acts on stationary objects
- DA force that always opposes motion
2.Which of the following is a non-contact force?
Easy
- AFriction
- BTension
- CGravitational force
- DNormal contact force
3.Which of the following is the correct equation for static friction?
Easy
- AF_f = μ_s F_N
- BF_f ≤ μ_s F_N
- CF_f ≥ μ_s F_N
- DF_f = μ_s / F_N
4.A box is sliding down a slope at constant speed. Which of the following is true about the frictional force acting on it?
Medium
- AIt acts up the slope and is equal in magnitude to the component of weight down the slope
- BIt acts down the slope and is equal in magnitude to the component of weight down the slope
- CIt acts up the slope and is greater than the component of weight down the slope
- DIt acts perpendicular to the slope
5.A spherical stone falls through the air and experiences a drag force of 3 mN at a particular instant. Air has a viscosity of 1.81 × 10^-5 Pa s and the stone has a radius of 0.04 m. Calculate the speed of the stone at that instant.
Medium
- A0.66 m/s
- B0.33 m/s
- C1.32 m/s
- D0.22 m/s
6.Which of the following are examples of contact forces? (Select all that apply)
Medium
- AFriction
- BTension
- CGravitational force
- DNormal contact force
- EMagnetic force
7.Which of the following statements about Newton's third law are correct? (Select all that apply)
Medium
- AThe forces are the same type
- BThe forces act on the same object
- CThe forces are equal in magnitude
- DThe forces act in opposite directions
- EThe forces act on different objects
8.Newton's first law states that a body will remain at rest or move with constant velocity unless acted on by a resultant force.
Easy
True or false?

もっと問題

9.The coefficient of static friction is always greater than the coefficient of dynamic friction for a given pair of surfaces.
Easy
True or false?
10.Match each force with its correct description.
Medium
- Weight
- Tension
- Normal contact force
- Friction
- Acts towards the surface of the planet
- Acts away from the mass along a rope or cable
- Acts perpendicular to a surface
- Acts in the opposite direction to motion
11.A car is moving at a constant velocity of 20 m/s on a straight road. What is the resultant force acting on the car?
Easy
- A0 N
- B20 N
- C200 N
- DIt cannot be determined without knowing the mass
12.Which of the following are properties of a Newton's third law force pair? (Select all that apply)
Medium
- AThey are the same type of force
- BThey act on the same object
- CThey are equal in magnitude
- DThey act in opposite directions
- EThey act on different objects
13.If the resultant force acting on an object is zero, the object must be at rest.
Easy
True or false?
14.Match each type of force with its correct description.
Medium
- Weight
- Normal contact force
- Tension
- Friction
- Acts towards the centre of the Earth
- Acts perpendicular to a surface
- Acts away from the mass along a string or rope
- Opposes motion between two surfaces in contact
15.A block is pushed with an increasing horizontal force until it just starts to move, then continues to slide. Order the stages of the frictional force acting on the block from first to last.
Medium
- Static friction increases as the applied force increases
- Static friction reaches its maximum value
- The block begins to move and dynamic friction acts
- Dynamic friction remains constant as the block slides
16.A book is at rest on a horizontal table. Which of the following correctly identifies the forces acting on the book?
Medium
- AWeight and normal contact force
- BWeight and tension
- CNormal contact force and friction
- DWeight, normal contact force and friction
17.A skier is being towed at constant speed across level snow. Which of the following statements are correct? (Select all that apply)
Medium
- AThe resultant force on the skier is zero
- BThe tension in the tow rope equals the frictional force
- CThe normal contact force equals the skier's weight
- DThe skier is accelerating because there is a forward force
- EThe frictional force is greater than the tension
18.A car of mass 1500 kg is travelling at 20 m/s. The brakes are applied and the car decelerates uniformly to rest in 5.0 s. What is the magnitude of the resultant force acting on the car during braking?
Medium
- A6000 N
- B300 N
- C1500 N
- D7500 N
19.A ball is thrown vertically upwards. Air resistance is not negligible. Which of the following statements are correct while the ball is moving upwards? (Select all that apply)
Hard
- AThe resultant force on the ball is downwards
- BThe acceleration of the ball is downwards
- CThe weight of the ball is greater than the air resistance
- DThe ball is in translational equilibrium
- EThe air resistance acts upwards
20.A toy sailboat has a weight of 30 N and is floating in water. It is pulled to the right with a horizontal force of 35 N and experiences a total resistive force of 5 N to the left. What is the magnitude and direction of the resultant force on the sailboat?
Medium
- A30 N to the right
- B30 N upwards
- C5 N to the left
- D70 N to the right
21.A helicopter is moving horizontally at constant velocity. The lift force from its blades is 25 kN vertically, and it experiences a horizontal air resistance force of 25 kN. What is the magnitude of the resultant force on the helicopter?
Medium
- A0 kN
- B25 kN
- C35 kN
- D50 kN
22.A rocket produces an upward thrust of 15 MN and has a weight of 8 MN. When in flight, the force due to air resistance is 500 kN. What is the resultant force on the rocket?
Medium
- A6.5 MN upwards
- B7.0 MN upwards
- C23.5 MN upwards
- D6.5 MN downwards
23.Three forces, 4 N, 8 N and 24 N, act on an object of mass 5 kg. Which of the following accelerations is NOT possible with any combination of these three forces?
Medium
- A1 m s⁻²
- B4 m s⁻²
- C7 m s⁻²
- D10 m s⁻²
24.A girl riding a skateboard increases her speed from 1 m s⁻¹ to 4 m s⁻¹ in 2.5 s. The force driving her forward is 72 N. What is the combined mass of the girl and the skateboard?
Medium
- A60 kg
- B45 kg
- C72 kg
- D30 kg
25.A car of mass 1250 kg travels at 0.5 m s⁻¹ towards a wall. The force–time graph of the collision shows a triangular pulse with a peak force of 2500 N lasting 0.5 s. What is the impulse of the collision?
Medium
- A625 N s
- B1250 N s
- C312.5 N s
- D2500 N s
26.A car of mass 1250 kg travels at 0.5 m s⁻¹ towards a wall. What is the momentum of the car before it collides with the wall?
Medium
- A625 kg m s⁻¹
- B1250 kg m s⁻¹
- C2500 kg m s⁻¹
- D0.0004 kg m s⁻¹
27.A stuntman of mass 85.0 kg falls and just before hitting the ground has a velocity of 9.08 m s⁻¹. After landing he is at rest. What is the change in momentum of the stuntman during landing?
Medium
- A772 kg m s⁻¹
- B0 kg m s⁻¹
- C85.0 kg m s⁻¹
- D9.08 kg m s⁻¹
28.A stuntman of mass 85.0 kg lands with an initial downward momentum of 772 kg m s⁻¹ and comes to rest. If he keeps his legs rigid, the impact time is 4.20 ms. What is the magnitude of the average resultant force on his legs?
Medium
- A1.84 × 10⁵ N
- B3.24 × 10³ N
- C772 N
- D1.84 × 10² N
29.A railway carriage C₁ of mass 1100 kg rolls at 6 m s⁻¹ towards a stationary carriage C₂ of mass 3300 kg. They collide and join together. What is the velocity of the joined carriages?
Medium
- A1.5 m s⁻¹
- B6.0 m s⁻¹
- C2.0 m s⁻¹
- D0.67 m s⁻¹
30.Two zorbs collide head-on. Zorb A (75 kg) moves at 2.0 m s⁻¹, zorb B (60 kg) moves at 1.5 m s⁻¹ in the opposite direction. After collision, zorb A moves at 0.8 m s⁻¹ and zorb B at 2.0 m s⁻¹ in opposite directions. What is the total kinetic energy before the collision?
Medium
- A217.5 J
- B150 J
- C300 J
- D435 J
31.Two zorbs collide head-on. Zorb A (75 kg) moves at 2.0 m s⁻¹, zorb B (60 kg) moves at 1.5 m s⁻¹ in the opposite direction. After collision, zorb A moves at 0.8 m s⁻¹ and zorb B at 2.0 m s⁻¹ in opposite directions. What is the total kinetic energy after the collision?
Medium
- A144 J
- B217.5 J
- C120 J
- D240 J
32.A skier is being towed at a constant speed across level snow. Which of the following statements are correct? (Select all that apply)
Medium
- AThe tension in the tow rope is equal in magnitude to the frictional force.
- BThe normal contact force equals the skier's weight.
- CThe resultant force on the skier is zero.
- DThe skier is accelerating because she is moving.
- EThe frictional force is greater than the tension.
33.Which of the following statements about a Newton's third-law force pair are correct? (Select all that apply)
Medium
- AThe two forces are always the same type of force.
- BThe two forces act on different objects.
- CThe two forces act on the same object.
- DThe two forces are equal in magnitude and opposite in direction.
- EThe two forces can be added to find the resultant force on one object.
34.In a collision where no external forces act, momentum is always conserved, but kinetic energy is only conserved in an elastic collision.
Medium
True or false?
35.A student is investigating static and dynamic friction by pulling a block with a force sensor. Which of the following statements about the experiment are correct? (Select all that apply)
Medium
- AThe maximum static friction force is reached just before the block starts to move.
- BOnce the block is moving at constant speed, the dynamic friction force is constant.
- CThe coefficient of static friction is typically smaller than the coefficient of dynamic friction for the same surfaces.
- DThe normal reaction force can be found by measuring the weight of the block if the surface is horizontal.
- EThe frictional force always acts in the same direction as the applied force.
36.In an experiment to measure the viscosity of a fluid using Stokes' law, the drag force on a small sphere is directly proportional to the radius of the sphere.
Easy
True or false?
37.Match each experimental method with the quantity it is used to determine.
Medium
- Measuring the gradient of a force–extension graph (force on y-axis, extension on x-axis)
- Measuring the area under a force–time graph
- Measuring the gradient of a velocity–time graph
- Measuring the area under a velocity–time graph
- Spring constant k
- Impulse
- Acceleration
- Displacement
38.A student is investigating how the frictional force on a block changes as the applied force is gradually increased from zero until the block moves at constant speed. Order the stages of the investigation from first to last.
Medium
- The applied force is zero and the block is stationary.
- The applied force is increased but the block remains stationary; static friction increases to match it.
- The applied force reaches a maximum and the block is on the point of moving.
- The block starts to move and the frictional force drops to a lower, constant value (dynamic friction).
- The block moves at constant speed with the applied force equal to the dynamic friction force.
39.A student uses a force sensor to measure the force exerted on a cart during a collision. The force–time graph is a triangle with a peak force of 120 N and a base of 0.15 s. The cart has a mass of 2.0 kg and is initially at rest. What is the final velocity of the cart?
Hard
- A4.5 m/s
- B9.0 m/s
- C18 m/s
- D2.25 m/s
40.Which of the following is the correct definition of a non-contact force?
Easy
- AA force which acts at a distance, without any physical contact between bodies, due to the action of a field
- BA force which acts between objects that are physically touching
- CA force which acts only when two surfaces rub together
- DA force which acts only on charged particles
41.State the name of the force that acts perpendicular to a surface when an object is supported by it.
Easy
42.A block of mass 1.5 kg is pulled along a table at constant speed by a force of 5.8 N. Calculate the magnitude of the dynamic coefficient of friction between the block and the table. Use g = 9.81 m/s².
Medium
43.Complete the following statement about Hooke's law.
Easy
The extension of a material is directly proportional to the applied force up to the ____ of ____.
44.A car of mass 1200 kg is moving at a constant velocity of 15 m/s. Calculate the resultant force acting on the car. Give your answer in N.
Medium
45.Complete the statement of Newton's first law.
Easy
A body will remain at rest or move with constant velocity unless acted on by a ____ force.
46.Complete the equation for dynamic friction.
Medium
Ff = μd × ____
47.A car of mass 1200 kg is moving at a constant velocity of 15 m s⁻¹. Calculate the resultant force acting on the car. Give your answer in N.
Medium
48.A block of mass 1.5 kg is pulled along a table at constant speed by a force of 5.8 N. Calculate the coefficient of dynamic friction between the block and the table. Use g = 9.81 m s⁻².
Medium
49.A skier of mass 52 kg is stationary on grass. The coefficient of static friction between the skis and grass is 0.12. Calculate the minimum force needed from the rope to get the skier just on the point of moving. Use g = 9.81 m s⁻².
Medium
50.A ball of mass 350 kg is hanging stationary from a cable. Calculate the tension in the cable. Use g = 9.81 m s⁻².
Medium
51.A spherical stone of volume 2.7 × 10⁻⁴ m³ falls through air and experiences a drag force of 3 mN. Air has a viscosity of 1.81 × 10⁻⁵ Pa s. Calculate the speed of the stone at that instant. Give your answer in m s⁻¹.
Medium
52.A block of mass 1.5 kg is pulled along a table at constant speed by a force of 5.8 N. Calculate the coefficient of dynamic friction between the block and the table. Use g = 9.81 m/s². Give your answer to 2 significant figures.
Medium
53.State the quantities that are conserved in an elastic collision and in an inelastic collision. Give your answer in terms of momentum and kinetic energy.
Hard
54.A physics textbook is at rest on a lab bench. Student A says the free-body diagram of the book is an example of Newton's third law. Student B says it is an example of Newton's first law. State and explain who is correct.
Hard
55.State what is meant by translational equilibrium.
Hard
56.A car is moving at constant velocity. The driving force is 6 kN. State the value of the frictional force and explain your reasoning.
Hard
57.A ball is suspended from a cable and is in translational equilibrium. Outline what is meant by translational equilibrium.
Medium
58.A block of mass 1.5 kg is pulled along a table at constant speed by a force of 5.8 N. Calculate the magnitude of the dynamic coefficient of friction between the block and the table. Use g = 9.81 m/s².
Medium
- A1.39
- B-0.61
- C2.39
- D0.39
59.A car of mass 1200 kg is moving at a constant velocity of 15 m/s. Calculate the resultant force acting on the car. Give your answer in N.
Medium
- A1 N
- B0 N
- C2 N
- D-1 N
60.A car of mass 1200 kg is moving at a constant velocity of 15 m s⁻¹. Calculate the resultant force acting on the car. Give your answer in N.
Medium
- A-1 N
- B0 N
- C2 N
- D1 N
61.A block of mass 1.5 kg is pulled along a table at constant speed by a force of 5.8 N. Calculate the coefficient of dynamic friction between the block and the table. Use g = 9.81 m s⁻².
Medium
- A1.39
- B0.39
- C-0.61
- D2.39
62.A skier of mass 52 kg is stationary on grass. The coefficient of static friction between the skis and grass is 0.12. Calculate the minimum force needed from the rope to get the skier just on the point of moving. Use g = 9.81 m s⁻².
Medium
- A73.4 N
- B61.2 N
- C55.1 N
- D67.3 N
63.A ball of mass 350 kg is hanging stationary from a cable. Calculate the tension in the cable. Use g = 9.81 m s⁻².
Medium
- A4120.3 N
- B3776.9 N
- C3090.1 N
- D3433.5 N
64.A spherical stone of volume 2.7 × 10⁻⁴ m³ falls through air and experiences a drag force of 3 mN. Air has a viscosity of 1.81 × 10⁻⁵ Pa s. Calculate the speed of the stone at that instant. Give your answer in m s⁻¹.
Medium
- A1.66 m s⁻¹
- B-0.34 m s⁻¹
- C2.66 m s⁻¹
- D0.66 m s⁻¹
65.A block of mass 1.5 kg is pulled along a table at constant speed by a force of 5.8 N. Calculate the coefficient of dynamic friction between the block and the table. Use g = 9.81 m/s². Give your answer to 2 significant figures.
Medium
- A1.39
- B2.39
- C0.39
- D-0.61

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