M8 Forces and Newton's laws — 9MA0 spec coverage

M8.1

Understand the concept of a force; understand and use Newton's first law.

A force is an interaction with magnitude and direction; common modelled forces include weight, normal reaction, tension, thrust, compression and resistance.
Draw a force diagram for the chosen body, including only forces acting on that body, then resolve forces along useful perpendicular directions.
Newton's first law says that a particle remains at rest or moves with constant velocity when the resultant force on it is zero.
A common error is to infer that a moving particle must have a forward resultant force; constant non-zero velocity also means zero resultant force.

Tier 1 · Easy

2 marks

ORIGINAL

A book rests on a horizontal table. Its weight is $35\,\text{N}$ . State the magnitude and direction of the normal reaction on the book and the resultant force on it.

Tier 2 · Standard

3 marks

ORIGINAL

A powered trolley moves in a straight line with constant velocity. Its motor exerts a forward force of $8\,\text{N}$ . Find the resistance force and explain your answer.

Tier 3 · Hard

4 marks

ORIGINAL

A particle moves with constant velocity under three coplanar forces. Two of the forces are $(6\mathbf{i}-8\mathbf{j})\,\text{N}$ and $(-2\mathbf{i}+5\mathbf{j})\,\text{N}$ . Find the third force, giving also its magnitude.

M8.2

Understand and use Newton's second law for motion in a straight line (forces in two perpendicular directions or simple 2-D vectors); extend to situations where forces need to be resolved (restricted to 2 dimensions).

Newton's second law is $\sum\mathbf{F}=m\mathbf{a}$ ; in two dimensions it gives one scalar equation in each resolved direction.
Choose axes parallel and perpendicular to the motion where possible, resolve every force onto those axes, then apply $\sum F=ma$ separately.
For motion along a fixed smooth plane, perpendicular acceleration is zero, so the perpendicular force equation determines the normal reaction while the parallel equation determines acceleration.
A common error is to write $F=ma$ for one force instead of the resultant, or to reverse signs for only some components after choosing a positive direction.

Tier 1 · Easy

2 marks

ORIGINAL

A constant resultant force of $12\,\text{N}$ acts on a particle of mass $3\,\text{kg}$ . Find its acceleration.

Tier 2 · Standard

4 marks

ORIGINAL

A $5\,\text{kg}$ particle is on a smooth plane inclined at $30^\circ$ to the horizontal. A force of $40\,\text{N}$ pulls it up the line of greatest slope. Using $g=9.8\,\text{m s}^{-2}$ , find its acceleration.

Tier 3 · Hard

7 marks

ORIGINAL

A $6\,\text{kg}$ particle is pulled up a smooth plane inclined at $20^\circ$ to the horizontal by a force of $50\,\text{N}$ acting at $15^\circ$ above the plane. Using $g=9.8\,\text{m s}^{-2}$ , find the normal reaction and the particle's acceleration up the plane.

M8.3

Understand and use weight and motion in a straight line under gravity; gravitational acceleration, g, and its value in S.I. units to varying degrees of accuracy.

Weight is the gravitational force $W=mg$ , directed vertically downwards; mass is measured in kilograms and does not depend on location.
The value of $g$ is not universal and depends on location; use the stated value, or the Edexcel default $9.8\,\text{m s}^{-2}$ when none is supplied.
For free motion near Earth's surface in the constant- $g$ model, acceleration is $g$ downwards, but a support or tension changes the resultant acceleration.
A common error is to call mass a force or to assume that the normal reaction always equals weight when the body has vertical acceleration.

Tier 1 · Easy

2 marks

ORIGINAL

Find the weight of a $2.5\,\text{kg}$ particle, using $g=9.8\,\text{m s}^{-2}$ .

Tier 2 · Standard

4 marks

ORIGINAL

A particle is released from rest and falls freely through $19.6\,\text{m}$ . Using $g=9.8\,\text{m s}^{-2}$ , find the time taken and its speed after falling this distance.

Tier 3 · Hard

4 marks

ORIGINAL

A person of mass $70\,\text{kg}$ stands on a scale in a lift. The scale exerts an upward force of $770\,\text{N}$ on the person. Using $g=9.8\,\text{m s}^{-2}$ , find the magnitude and direction of the lift's acceleration.

M8.4

Understand and use Newton's third law; equilibrium of forces on a particle and motion in a straight line; apply to smooth pulleys and connected particles; resolve forces in 2 dimensions; equilibrium of a particle under coplanar forces.

Newton's third-law forces are equal and opposite, act on different bodies and arise from the same interaction; they therefore do not cancel on one body's force diagram.
For a particle in equilibrium under coplanar forces, resolve in two independent directions and set both component resultants equal to zero.
For connected particles, draw separate force diagrams, use a common acceleration magnitude while the string is taut, and use one tension for a light string over a smooth pulley before solving simultaneous $F=ma$ equations.
A common error is to treat weight and normal reaction as a third-law pair; both act on the same body, whereas a third-law pair acts on different bodies.

Tier 1 · Easy

2 marks

ORIGINAL

A book pushes down on a table with force $P$ . State the Newton's third-law partner to this force.

Tier 2 · Standard

5 marks

ORIGINAL

A particle is in equilibrium under three coplanar forces. One force is $14\,\text{N}$ east and another is $10\,\text{N}$ at $120^\circ$ anticlockwise from east. Find the third force as a vector in east-north components and find its magnitude.

Tier 3 · Hard

7 marks

ORIGINAL

A $4\,\text{kg}$ particle on a smooth plane inclined at $30^\circ$ is connected by a light inextensible string over a smooth pulley to a freely hanging $3\,\text{kg}$ particle. Using $g=9.8\,\text{m s}^{-2}$ , find the acceleration of the system and the tension in the string.

M8.5

Understand and use addition of forces; resultant forces; dynamics for motion in a plane.

Forces add as vectors, so a resultant may be written in component form and then converted to magnitude-direction form.
Resolve each force along two perpendicular axes, add corresponding components, and apply $\sum\mathbf{F}=m\mathbf{a}$ to obtain the acceleration vector.
If the resultant is $X\mathbf{i}+Y\mathbf{j}$ , its magnitude is $\sqrt{X^2+Y^2}$ and its direction must be placed in the correct quadrant from the component signs.
A common error is to add force magnitudes without accounting for their directions, or to quote an inverse-tangent angle in the wrong quadrant.

Tier 1 · Easy

3 marks

ORIGINAL

A force is $(6\mathbf{i}-8\mathbf{j})\,\text{N}$ . Find its magnitude and its angle below the positive $\mathbf{i}$ direction.

Tier 2 · Standard

5 marks

ORIGINAL

Two forces of magnitudes $12\,\text{N}$ and $10\,\text{N}$ act at an angle of $120^\circ$ to each other. Find the magnitude of their resultant and the angle the resultant makes with the $12\,\text{N}$ force.

Tier 3 · Hard

7 marks

ORIGINAL

A particle of mass $5\,\text{kg}$ has initial velocity $(2\mathbf{i}-\mathbf{j})\,\text{m s}^{-1}$ . Constant forces $(15\mathbf{i}+20\mathbf{j})\,\text{N}$ and $(-5\mathbf{i}+10\mathbf{j})\,\text{N}$ act on it. Find its acceleration, velocity after $3\,\text{s}$ and displacement during those $3\,\text{s}$ .

M8.6

Understand and use the F ≤ μR model for friction; coefficient of friction; motion of a body on a rough surface; limiting friction and statics.

Friction acts to oppose actual or impending relative motion; in equilibrium its magnitude adjusts within $0\leq F\leq\mu R$ .
Find the normal reaction first, decide the likely direction of motion, and use $F=\mu R$ only when friction is limiting or the particle is moving in this model.
For a range of equilibrium values, write the required friction in terms of the applied force and impose $|F|\leq\mu R$ before solving the resulting inequality.
A common error is to set $F=\mu R$ in every static problem; away from limiting equilibrium, friction may be strictly smaller than $\mu R$ .

Tier 1 · Easy

3 marks

ORIGINAL

A block is in equilibrium on a rough horizontal surface. The normal reaction is $80\,\text{N}$ and the coefficient of friction is $0.30$ . A horizontal force of $10\,\text{N}$ acts on the block. Find the friction force and the greatest possible friction force.

Tier 2 · Standard

5 marks

ORIGINAL

A $6\,\text{kg}$ block is moving on a rough horizontal surface with coefficient of friction $0.25$ . A horizontal force of $20\,\text{N}$ acts in the direction of motion. Using $g=9.8\,\text{m s}^{-2}$ , find its acceleration.

Tier 3 · Hard

7 marks

ORIGINAL

A $10\,\text{kg}$ particle rests on a rough plane inclined at $20^\circ$ to the horizontal. The coefficient of friction is $0.30$ . A force of magnitude $P\,\text{N}$ acts up the plane. Using $g=9.8\,\text{m s}^{-2}$ , find the complete range of values of $P$ for which the particle remains in equilibrium.