describe and use the concept of weight as the effect of a gravitational field on a mass and recall that the weight of an object is equal to the product of its mass and the acceleration of free fall

Weight – the Effect of a Gravitational Field on a Mass

Learning Objective

Describe and use the concept of weight as the effect of a gravitational field on a mass, and recall that the weight of an object is equal to the product of its mass and the acceleration of free fall.

1. Physical Quantities & Units

• Mass (m) – amount of matter, intrinsic to the object. Unit: kilogram (kg).
• Weight (W) – gravitational force acting on a mass. Unit: newton (N).
• Acceleration due to gravity (g) – rate at which objects accelerate toward the centre of the Earth. Near the surface, g ≈ 9.81 m s⁻² (value varies with location).

2. Scalars & Vectors

• Weight is a vector – it has magnitude and a direction (toward the centre of the Earth).
• Mass is a scalar – it has magnitude only.

3. Weight as a Gravitational Force

The weight of an object is the gravitational force acting on its mass:

\[

\boxed{W = mg}

\]

In vector form (taking upward as the positive \(\hat{\mathbf{j}}\) direction):

\[

\mathbf{W} = -\,mg\,\hat{\mathbf{j}}

\]

4. Direction of Weight & Free‑Body Diagrams

Weight always acts vertically downwards, i.e. toward the centre of the Earth. In a free‑body diagram it is drawn as a straight arrow pointing down.

Example – Book on a Table

• \(\mathbf{W}\) – weight, downwards.
• \(\mathbf{N}\) – normal reaction, upwards.

Since the book is at rest, the net force is zero (Newton’s First Law):

\[

\mathbf{N} + \mathbf{W} = \mathbf{0}

\]

5. Weight‑Only Dynamics (AO1 + AO2)

Applying Newton’s second law when weight is the only external force.

1. Free fall (no air resistance)

   \[

   \mathbf{F}_{\text{net}} = \mathbf{W} = mg \;\Rightarrow\; a = g

   \]

   The object accelerates downward at \(g\).

2. Equilibrium on a vertical rope

   Tension \(T\) balances weight: \(T = mg\).

3. Inclined plane – resolve weight into components:

   \[

   W{\parallel}=mg\sin\theta,\qquad W{\perp}=mg\cos\theta

   \]

   Where \(\theta\) is the angle of the plane with the horizontal.

4. Example problem – A 2 kg block rests on a 30° incline (smooth).

   • \(W_{\parallel}=2(9.81)\sin30^{\circ}=9.81\;\text{N}\) down the slope.
   • \(W_{\perp}=2(9.81)\cos30^{\circ}=16.97\;\text{N}\) into the plane.

   If the block is held by a string parallel to the slope, the tension required is \(T = 9.81\;\text{N}\) upward along the plane.

6. Kinematics of a Falling Body (AO1)

From \(\mathbf{F}=mg\) we obtain constant acceleration \(a=g\). The standard equations of motion apply:

Worked example – A stone is dropped from a cliff 20 m high. Find the time to reach the ground and the impact speed.

• Initial velocity \(u=0\). Use \(s = \tfrac12 gt^{2}\) → \(20 = \tfrac12(9.81)t^{2}\) → \(t = 2.02\;\text{s}\).
• Impact speed \(v = gt = 9.81(2.02) = 19.8\;\text{m s}^{-1}\) (downwards).

7. Newton’s Laws (Re‑linked to Weight)

• First Law (Inertia) – Motion continues unchanged unless acted upon by a net external force (e.g., weight).
• Second Law – \(\mathbf{F}{\text{net}} = m\mathbf{a}\). For a weight‑only situation, \(\mathbf{F}{\text{net}} = \mathbf{W}\) giving \(a = g\).
• Third Law – The Earth exerts a downward force on the object (its weight) and the object exerts an equal upward reaction on the Earth.

8. Momentum – Definition, Conservation & Connection to Weight

Definition: \(\mathbf{p}=m\mathbf{v}\) (vector).

Second law in momentum form: \(\displaystyle\frac{d\mathbf{p}}{dt}= \mathbf{F}_{\text{net}}\).

When weight is the only vertical force, \(\displaystyle\frac{dp_{y}}{dt}= -mg\).

Conservation principle: If \(\mathbf{F}_{\text{net}}=0\) (e.g., in horizontal motion on a frictionless surface), total momentum is constant.

9. Density, Pressure & Buoyancy (AO1 – 4.3)

Key definitions

• Density (\(\rho\)) – mass per unit volume, \(\rho = \dfrac{m}{V}\) (kg m⁻³).
• Pressure (p) – force per unit area, \(p = \dfrac{F}{A}\) (Pa = N m⁻²).

Hydrostatic pressure

For a fluid of uniform density, pressure increases with depth:

\[

\boxed{\Delta p = \rho g \Delta h}

\]

\(\Delta h\) is the vertical depth, \(g\) the local acceleration due to gravity.

Up‑thrust (Buoyant force)

The weight of the displaced fluid produces an upward force:

\[

\boxed{F{\text{B}} = \rho{\text{fluid}}\,g\,V_{\text{disp}}}

\]

where \(V_{\text{disp}}\) is the volume of fluid displaced.

Worked example – Buoyancy of a wooden block

• Block dimensions: \(0.10\;\text{m} \times 0.10\;\text{m} \times 0.20\;\text{m}\) → \(V = 2.0\times10^{-3}\;\text{m}^{3}\).
• Density of wood: \(\rho_{\text{wood}} = 600\;\text{kg m}^{-3}\). Mass \(m = \rho V = 1.2\;\text{kg}\).
• Weight: \(W = mg = 1.2(9.81)=11.8\;\text{N}\) (downwards).
• Water density: \(\rho_{\text{water}} = 1000\;\text{kg m}^{-3}\).

  Up‑thrust \(F_{\text{B}} = 1000(9.81)(2.0\times10^{-3}) = 19.6\;\text{N}\) (upwards).

• Since \(F_{\text{B}} > W\), the block floats with part of its volume above the surface.

10. Gravitational Potential Energy (AO1 + AO2)

Work done against weight to raise a mass through a vertical height \(h\):

\[

\boxed{E_{\text{p}} = mgh}

\]

Units: joule (J) = N m.

Example – Lifting a 5 kg sack 3 m

\[

E_{\text{p}} = (5)(9.81)(3) = 147.2\;\text{J}

\]

11. Weight vs. Mass – Common Misconceptions

1. Mass is an intrinsic property; weight depends on the local value of \(g\).
2. On the Moon (\(g_{\text{Moon}} \approx 1.62\;\text{m s}^{-2}\)) an object’s weight is about one‑sixth of its Earth weight, but its mass is unchanged.
3. Weight is a force (N), not a mass (kg).

12. Practical Examples

13. Summary Checklist (AO1)

• Weight is a vector: \(\mathbf{W} = -mg\hat{\mathbf{j}}\).
• Magnitude: \(W = mg\) (units N).
• Mass is invariant; weight varies with \(g\).
• Free‑fall acceleration \(a = g\); kinematic equations follow directly.
• Resolve weight into components on inclined planes or in equilibrium problems.
• Newton’s second law links weight to acceleration and to the rate of change of momentum.
• Hydrostatic pressure \(\Delta p = \rho g\Delta h\) and buoyant force \(F_{\text{B}} = \rho g V\) arise from the weight of fluid columns.
• Gravitational potential energy \(E_{\text{p}} = mgh\).
• Momentum \(\mathbf{p}=m\mathbf{v}\) is conserved when the net external force (including weight) is zero.