Topic 2: Mechanics

1. Equations for uniform acceleration

• There are three basic equations of motion for bodies moving with uniform acceleration.
• These equations relate initial velocity, final velocity, acceleration, time and distance covered by a moving body.
• To simplify the derivation of these equations, we assume that the motion is along a straight line.
• Hence, we consider only the magnitude of displacements, velocities, and acceleration.
• •  Consider a body moving with initial velocity [math]\vec{v_i}[/math] in a straight line with uniform acceleration [math]\vec{a}[/math].
• Its velocity becomes [math]\vec{v}[/math] after time t.
• The motion of body is described by speed-time graph as shown in figure 5 by line AB.
• The slope of line AB is acceleration [math]\vec{a}[/math].
• The total distance covered by the body is shown by the shaded area under the line AB.
• Equations of motion can be obtained easily from this graph.

• First equation of motion
• Speed-time graph for the motion of a body is shown in figure 5. Slope of line AB gives the acceleration a of a body.

  
  Figure 1 Speed-time graph. Area under the graph gives the distance covered by the body.

• 
• Second equation of motion
• In speed-time graph shown in figure 5, the total distance S travelled by the body is equal to the total area OABD under the graph. That is
• Total distance(S)=Area of (Rectangle OACD = triangle ACB)     (2)
• 
• Third equation of motion
• The whole area OABD under the speed-time graph in figure 5 gives the total distance S traversed by the body.
• [math] \text{Total area } OABD = S = \frac{(OA + BD)}{2} * OD [/math]
• [math]2S = (OA + BD) * OD [/math]
• [math] \text{Multiply by } \frac{BC}{OD} \text{ on both sides:} [/math]
• [math]2S * \frac{BC}{OD} = (OA + BD)* OD * \frac{BC}{OD} [/math]
• Then
• [math] 2S * \frac{BC}{OD} = (OA + BD) * BC \qquad (4) [/math]
• We know: [math] \frac{BC}{OD} = \vec{a}, \, OA = \vec{v_i}, \, BD = \vec{v_f}, \, BC = BD – CD, \, \text{and } CD = OA = \vec{v_i} [/math]
• So, equation (4) becomes:
• [math]2S * \frac{BC}{OD} = (OA + BD) * (BD – CD) \\
  2\vec{a} S = (\vec{v_i} + \vec{v_f}) * (\vec{v_f} – \vec{v_i}) [/math]
• [math]2\vec{a} S = \vec{v_f}^2 – \vec{v_i}^2 \qquad (5)  [/math]

2. Motion in a straight line

• Felix Baumgartner broke the records for the highest altitude reached by a parachute jump.
• The highest manned balloon flight, and the greatest free fall velocity in October 2012.
• At a height of over 39 km, he leaped and attained a speed of [math]377 m.s^{-1}[/math] ([math]1357 km.h^{-1}[/math]).
• Careful planning and design are necessary for such a challenge to be successfully completed.
• Felix relied on physicists’ calculations using the equations of motion to predict his time of fall and to determine when it was safe to open his pressurized space suit.
• Felix also needed a balloon inflated to precisely the right pressure at sea level so that it could expand as the atmospheric pressure decreased at high altitude.
• Distance and Displacement
  • Distance: The total length of the path an object follows.
    Unit: meter
    Formula: Distance = speed * time
    Quantity: scalar
  • Displacement: The distance an object travels from its initial to its final position
    Unit: meter
    Formula:  [math] \vec{s} = \vec{s_f} \, \text{(final point)} – \vec{s_i} \, \text{(initial point)} [/math]
    Quantity: Vector
• Speed and velocity:
  • Speed: Speed is a scalar quantity that refers to how fast an object is moving.
    It is defined as the distance traveled per unit time (e.g., meters per second, miles per hour).
    Speed is a rate of motion, and it has no direction.
  • Velocity: Velocity is a vector quantity that refers to an object’s speed in a specific direction.
    It is defined as the displacement (change in position) per unit time.
    Velocity has both magnitude (amount of movement) and direction.
  • Acceleration: Acceleration is a vector quantity that measures the rate of change of velocity.
    It is defined as the change in velocity (Δv) per unit time (t).
    Acceleration is a change in the speed or direction of an object.
  • Types of acceleration:
    1. Uniform acceleration: When the acceleration is constant.
    2. Non-uniform acceleration: When the acceleration changes over time.
    3. Instantaneous acceleration: The acceleration at a specific instant.

⇒ Displacement-time graphs

• The terms distance and displacement are used interchangeably when the motion is in a straight line.

  
  Figure 2 Displacement-time graph for a walker

• Similarly, if the motion is in a straight line, then speed and velocity are also used interchangeably.
• In a displacement-time graph, time is taken along horizontal axis while vertical axis shows the distance covered by the object.
• • A walker’s displacement–time (s–t) graph is displayed in Figure 1.
• She advances to the right for the first three seconds, which is selected to be positive.
• After that, she stops moving for two seconds, maintaining her displacement.
• Finally, she moves back to the left for five seconds, ending up where she began.
• She has therefore moved zero after 10 seconds.
• • The graph’s gradient may be used to determine the walker’s speed.
• Over the region AB:

[math] \text{Velocity} = \frac{\text{Change in displacement}}{\text{time taken}} \\
v = \frac{\text{final displacement} – \text{initial displacement}}{\text{time taken}} \\
v = \frac{(10 – 0) \, \text{m}}{3 \, \text{s}} \\
v = \frac{10 \, \text{m}}{3 \, \text{s}} \\
v = 3.3 \, \text{m/s} [/math]

• Over the region BC:
  No displacement cover in that region. So,
  v = 0
• Over the region CD:
• [math]v = \frac{\text{final displacement} – \text{initial displacement}}{\text{time taken}} \\
  v = \frac{(0 – 10)}{5} \\
  v = -2 \, \text{m/s} [/math]
• Because the gradient is negative over the region CD the velocity is negative.

  
  Figure 3 A displacement–time graph for a vehicle that is accelerating.

• There are three constant velocities shown in Figure 3. This indicates that the graphs are “straight line” since the graphs’ gradients are constant.
• A displacement–time graph of an accelerating vehicle is shown in Figure 3. The gradient rises with increasing velocity.
• We measure the gradient at point P by drawing a tangent to the curve there in order to get the velocity. We quantify a tiny displacement shift, Δ [math] \vec{s}  [/math] , that happened during a little time span, Δt.

⇒ Velocity–time graphs

• In a Velocity-time graph, time is taken along x-axis and velocity is taken along y-axis.
• Independent values along x-axis while dependent values along y-axis.
  • An airplane’s velocity–time graph during takeoff is seen in Figure 3.
• Part AB of the graph illustrates how the plane accelerates initially at a constant pace.

  
  Figure 4 airplane’s velocity–time graph during takeoff

• After then, the plane accelerates more slowly until it reaches point D, where its velocity is constant.
• The graph illustrates that the acceleration is 4 m/s across portion AB.
• The graph’s gradient is as follows: ([math]80ms^{-1}[/math])/(20s).
• However, the gradient is altering throughout the region BC, thus we now use the following formula:
• 
• Where Δ [math] \vec{v} [/math] means a small change in velocity, and Δt means a small interval of time. At point P:
• 
• Figure 5 shows how we can use a velocity–time graph to calculate the distance travelled by a motorbike.
• In this graph, the motorbike travels at a constant velocity over the period AB, before decelerating over the period BC. While the velocity is constant, the distance travelled is represented by area A1, which is 320m.
• While the bike decelerates, we could use the formula
• 
• So

  
  Figure 5 How to use a velocity–time graph to calculate the distance travelled by a motorbike.
  

• displacement = average velocity * time taken
• But the average velocity is 10ms−1, which is the average of [math] 0ms^{-1}[/math] and [math]20ms^{-1}[/math].
• So, the distance can also be calculated using the area:
• 
• Figure 5 also shows that when the gradient of a velocity–time graph is zero, then the velocity is constant.
• When the gradient is negative, the bike is decelerating.
• The area under a velocity–time graph is the distance travelled.
• The gradient of a velocity–time graph is the acceleration.

3. Scalar and vector quantities

⇒ Scalar Quantity

• A scalar quantity is a physical quantity that has only magnitude (amount) but no direction.
• Scalars are used to describe quantities that are invariant under coordinate transformations, meaning their value remains the same regardless of the coordinate system used to measure them.
• Examples of scalar quantities are
  • Temperature (represented by T, (SI unit is Kelvin (, Base quantity)
  • Time (represented by t, (SI unit is second (s), Base quantity)
  • Mass (represented by m, (SI unit is kilogram (kg), Base quantity)
  • Length (represented by l (SI unit is meter (m), Base quantity)
  • Electric current (represented by I (SI unit is ampere (A), Base quantity)
  • Intensity of light (represented by L (SI unit is candela (cd), Base quantity)
  • Amount of a substance (represented by n (SI unit is mole (mol), Base quantity)
  • Density (represented by , (SI unit is kilogram per meter cube ([math]\text{kgm}^{-3} [/math]), Drive Quantity)
  • Pressure (represented by V (SI unit is meter cube ([math] \text{m}^3 [/math]), Drive quantity)
  • Charge (represented by Q (SI unit is coulomb (C or As), Drive quantity)
• Scalar quantities are often represented by a single number or a simple value, unlike vectors, which have both magnitude and direction.
• Some key properties of scalar quantities are
  • Commutativity: Scalars can be added or multiplied in any order.
  • Associativity: Scalars can be grouped in any way when adding or multiplying.
  • Distribution: Scalars can be distributed over addition and multiplication.

⇒ Vector Quantity

• A vector quantity is a physical quantity that has both magnitude (amount) and direction.
• Vectors are used to describe quantities that have both size and direction, and are often represented graphically as arrows in a coordinate system.

Representation

Figure 6 A vector representation

• Examples of vector quantities include:
  • Displacement (represented by d (SI unit is meter (m), Base Quantity)
  • Velocity (illustrated by v (SI unit is meter per second ([math]\text{m.s}^{-1}[/math]), Drive quantity)
  • Acceleration (determined by a (SI unit is meter per second per second ([math] \text{m.s}^{-2} [/math]), Drive quantity)
  • Force, Weight (showed by F / W (same SI unit is newton (N (, Drive quantity)
• Vector quantities are often represented mathematically using boldface letters.
• Some key properties of vector quantities:
  • Magnitude (length): The size of the vector.
  • Direction: The direction in which the vector points.
  • Addition: Vectors can be added graphically or mathematically.
  • Scalar multiplication: Vectors can be multiplied by a number, which changes their magnitude.
  • Dot product (scalar product): The product of two vectors that results in a scalar.
  • Cross product (vector product): The product of two vectors that results in another vector.

⇒ Resultant Vector

• The resultant vector, also known as the net vector or total vector, is the vector that results from the combination of two or more vectors.
• It is the vector that represents the overall effect of all the vectors acting together.

  
  Figure 7 R represent the resultant vector

• Properties of resultant vectors:
  • Magnitude: The length of the resultant vector is the sum of the magnitudes of the individual vectors.
  • Direction: The direction of the resultant vector is the angle between the individual vectors.
  • Association: The order in which the vectors are added does not affect the resultant vector.
  • Commutativity: The resultant vector is the same when the individual vectors are added in reverse order.
• Resultant vectors are used in various applications, such as:
  • Force vectors: To find the net force acting on an object.
  • Velocity vectors: To find the net velocity of an object.
  • Acceleration vectors: To find the net acceleration of an object.

4. The Addition of vectors

• The addition of vectors is a fundamental operation in vector mathematics.
• It involves combining two or more vectors to produce a resultant vector. Here are the key aspects of vector addition:

  
  Figure 8 Addition of vectors

  • Graphical Method: Vectors are added graphically by connecting the tail of one vector ([math] \vec{b} [/math]) to the head of another vector ([math] \vec{a} [/math]). The resultant vector ([math] \vec{AC} [/math]) is the vector that connects the tail of the first vector to the head of the last vector.
  • Mathematical Method: Vectors can be added mathematically using the component method or the unit vector method.
  • Component Method: Vectors are added component-wise, meaning that the corresponding components ([math] \vec{a}, \vec{b} [/math]) of the vectors are added together.
  • Unit Vector Method: Vectors are expressed in terms of unit vectors (i, j, k) and then added component-wise.
• Types of Vector Addition:
• Scalar addition: Adding a scalar to a vector.
• Vector addition: Adding two or more vectors.
• You are expected to be able to calculate vector magnitudes.
• When two vectors are at right angles.
• But the calculations are harder when two vectors are separated by a different angle.
• Example 1
• A rambler walks a distance of 8km travelling due east, before walking 6km due north. Calculate their displacement.

  
  Figure 9 graphical representation of rambler walk

  • Solution

    Displacement is a vector quantity, so we must calculate its magnitude and direction. This calculation, for vectors at right angles to each other, can be done using Pythagoras’ theorem and the laws of trigonometry.

  • [math] |AC|^2 = |AB|^2 + |BC|^2 \\
    |AC|^2 = |8 \, \text{km}|^2 + |6 \, \text{km}|^2 \\
    |AC|^2 = |64 + 36| * \text{km}^2 \\
    |AC|^2 = 100 \, \text{km}^2 \\
    \text{Taking the square root on both sides:} \\
    \sqrt{|AC|^2} = \sqrt{100 \, \text{km}^2} \\
    \text{Then,} \, |AC| = 10 \, \text{km} [/math]
  • Directions can be calculated on a bearing from due north; this is the angle θ, in Figure 4, which is also the angle θ, in the triangle.
    The displacement is 10km on a bearing of 53°

5. The resolution of vectors

• Resolving a vector Split it into two mutually perpendicular components that add up to the original vector.

  
  Figure 10 a passenger pulling his wheelie bag

• Figure 10 shows a passenger pulling his wheelie bag at the airport. He pulls the bag with a force, F, part of which helps to pull the bag forward and part of which pulls the bag upwards, which is useful when the bag hits a step.
• The force, F, can be resolved into two components:
• A horizontal component F_h. 
• A vertical component F_v.
• A vector can be resolved into any two components that are perpendicular, but resolving a vector into vertical and horizontal components is often useful, due to the action of gravity.
• Using the laws of trigonometry:
• 
• The vertical component of the force is
• [math]F_v= F sinθ [/math]
• The horizontal component of the force is
• [math] F_h = F cosθ [/math]

⇒ An inclined plane

• An inclined plane is a flat surface that is tilted at an angle to the horizontal. It is a fundamental concept in physics.
• An inclined plane is used to describe a variety of phenomena, such as
  • Motion on an inclined plane: Objects can move up or down an inclined plane, and the force of gravity acts along the surface of the plane.
  • Forces on an inclined plane: The force of gravity, normal force, and frictional force act on an object on an inclined plane.
  • Inclined plane in mechanics: Inclined planes are used to change the direction of motion, increase or decrease the force of gravity, and to create a mechanical advantage.
  • Real-world applications: Inclined planes are used in various real-world applications, such as:
  • Ramps: Used to load and unload heavy objects.
  • Stairs: A series of inclined planes that connect different levels.
  • Roofs: Inclined planes that provide protection from the elements.
  • Highways: Inclined planes that connect different elevations.
  • Figure 6 shows a car at rest on a sloping road. The weight of the car acts vertically downwards, but here it is useful to resolve the weight in directions parallel (||) and perpendicular (⊥) to the road.

    
    Figure 11 a car at rest on a sloping road.

  • The component of the weight parallel to the road provides a force to accelerate the car downhill.
  • The component of the weight acting along the slope is
  • The component of the weight acting perpendicular to the slope is
  • W _||=W sinθ
  • The component of the weight acting perpendicular to the slope is
  • W _⊥=W cosθ

6. Projectile motion – or falling sideways

• Projectile motion is a fundamental concept in physics that describes the motion of an object that is thrown, launched, or projected into the air.
• It’s a two-dimensional motion, meaning the object moves in both horizontal and vertical directions.
• the key aspects of projectile motion:
  1. Trajectory: The path the object follows under the influence of gravity.
  2. Initial velocity (v_i): The velocity at which the object is launched.
  3. Angle of launch (θ): The angle at which the object is launched, relative to the horizontal.
  4. Gravity (g): The acceleration due to gravity, which pulls the object downwards.
  5. Time of flight (T): The total time the object is in the air.
  6. Range (R): The horizontal distance the object travels.
  7. Maximum height (H): The highest point the object reaches.
• According to Newton’s first Law of motion, unless an unbalanced force acts upon an item, it will continue to travel in a straight path at a constant speed or remain at rest.
• A resultant force applied on an object causes it to accelerate in the direction stated by Newton’s second law of motion.
• Force = mass * acceleration
• Projectile paths may be predicted using the equations of motion.
• A vector velocity may be resolved into horizontal and vertical components.

⇒ Falling sideways

• A diver may be seen leaping into the water in Figure 12.
• Every picture is captured 0.2 seconds apart.

  
  Figure 12 A man jumping off a diving board into the sea

• It is evident that the diver’s sideways displacement is consistent across all photo frames.
• This is because he is not being affected by any horizontal force, therefore he continues to go in that direction at a steady speed.
• However, his downward displacement gets larger with every image frame.
• This is because he is falling faster due to the acceleration caused by gravity.
• The diver’s ability to move both vertically and horizontally independently of one another is a crucial concept.
• Since the diver has no beginning velocity in a vertical direction, he will always reach the water at the same moment when he jumps horizontally from the diving board.
• But the quicker he sprints sideways, the farther his fall will take him from the board.
• In all of the calculations that follow, we shall assume that we may ignore the effects of air resistance.
• However, when something is moving very quickly.
• A golf ball for example
  • The effects of air resistance need to be taken into account.
  • The effect of air resistance on the path of a golf ball is to reduce its maximum height and to reduce the horizontal distance (range) that it travels.

7. Newton’s first law – free body diagrams:

• “An object at rest will remain at rest, and an object in motion will continue to move with a constant velocity, unless acted upon by an external force.“
• “If an object is at rest, it will stay at rest, unless a force is applied to it”.
• “If an object is moving, it will continue to move with a constant velocity (speed and direction), unless a force is applied to it”.
• This law applies to all objects, big or small, and is a fundamental concept in understanding how objects move and respond to forces.

  
  Figure 13 By newton’s law of motion

• Some key points to note about the Law of Inertia:
  – Inertia is a property of an object that describes its tendency to resist changes in its motion.
  – The law applies to all objects, regardless of their mass or size.
  – The law only applies to objects that are not subject to external forces. If a force is applied, the object’s motion will change.
  – The law is often referred to as the “law of inertia” because it describes the tendency of objects to maintain their state of motion.

⇒ Free body diagram:

• A free body diagram is a graphical representation of an object and the forces acting upon it. It’s a crucial tool in physics and engineering to visualize and analyze the forces that affect an object’s motion.
• A free body diagram typically includes:
  1. The object: Represented by a box, circle, or other shape.
  2. Forces: Arrows that represent the forces acting on the object, labeled with their magnitude and direction.
  3. Axes: Coordinate axes (x, y, and sometimes z) that help to define the orientation of the forces.

     
     Figure 14 When external force applies on a moving object then it will change its motion

     Types of forces that might be shown on a free body diagram:
     External forces:
     – Frictional force (f)
     – Normal force (N)
     – Applied force (F)
     – Gravity (g)

• Internal forces:
  – Tension (T)
  – Spring force (k)
  – Air resistance (D)
• Some benefits of using free body diagrams:
  – Help to clarify complex force systems
  – Simplify problem-solving
  – Enhance understanding of force interactions
  – Facilitate calculation of net forces and motion

⇒ Examples

(1)

• After Galileo made some discoveries on friction, Newton expanded on them.
• Galileo saw balls tumbling over various curves.
  
• Figure 15 Both (starting and ending) points are equal without incline plane
• The ball will run down one side of the curve then up the other.
• Galileo noticed that if smooth surfaces were used, the ball got closer to its original height (its height at the starting point).
• Galileo reasoned that the ball would get to the original height if there were no friction.
• The ball would get to the original height if there were no outside force (unbalanced force).
  
• Figure 16 both points (ending and starting) are same but one end has an incline plane
• In all cases, the ball will return to its starting height regardless of the angles.
• The ball will go a larger distance but still not reach its initial height if the second curve’s inclination is smaller than the firsts because of friction, or the ball meeting resistance from the surface it is racing along.
• Even though the two ramps had different slopes, in the absence of friction, the ball would roll up the opposing slope to its original height.
  
• Figure 17 Second end is not equal to first starting point
• Galileo came to the conclusion that if the curve terminated without an inclination, the ball would continue indefinitely till friction finally stopped it.
• These findings led Newton to the conclusion that an item may remain in motion without the assistance of a force. The ball is really stopped from going any farther by an outside force.

8. Newton’s second law of motion:

Newton’s Second Law, also known as the Law of Acceleration, relates the motion of an object to the force acting upon it. It states:

• “The acceleration of an object is directly proportional to the force applied and inversely proportional to its mass.”
  
• Where:
  – [math]\vec{F}[/math] is the net force acting on an object, unit is N (newton)
  – m is the mass of the object, unit is kg (kilogram)
  –  [math]\vec{a}[/math] is the acceleration of the object, unit is m/s (meter per second)
• This law means that:
  – The more force applied to an object, the more it will accelerate ([math]\vec{a} \propto \vec{F} [/math]) where the mass of an object will be constant.
  – The heavier an object is (more massive), the less it will accelerate when a force is applied([math]\vec{a} \propto \frac{1}{m}
  [/math]).

   

  
  Figure 19 Newton’s second Law very clearly example with respect to two different masses, different acceleration, and different force

• Some key aspects of Newton’s Second Law:
  – Force and acceleration are vectors, so they have both magnitude and direction.
  – Mass is a scalar quantity, so it has only magnitude (amount of matter).
  – The law applies to all objects, big or small, and is a fundamental principle in understanding how objects move and respond to forces.

⇒ Examples:
(1)

• Figure 9 shows the forces on a cyclist accelerating along the road. The forces in the vertical direction balance, but the force pushing her along the road, F, is greater than the drag forces, D, acting on her. The mass of the cyclist and the bicycle is 100kg. Calculate her acceleration.

  
  Figure 20 The forces on a cyclist

Given Data:
Drag Force = [math]\vec{D} = 180 N[/math]
Road force = [math]\vec{F} = 320N[/math]
The mass of the cyclist and the bicycle= 100kg
Find data:
Acceleration =?
Formula:

[math]\vec{F} – \vec{D} = m\vec{a}[/math]

Solution:
Her acceleration can be calculated as follows.

[math]\vec{F} – \vec{D} = m\vec{a}[/math]

Put values

[math]320 -180 =100 * \vec{a}[/math]
[math]\frac{140}{100} = \vec{a} [/math]
[math]\vec{a}=1.4m/s[/math]

9. Gravitational field strength:

• Gravitational field strength is a measure of the gravitational force exerted on an object per unit mass at a given point in space.
• It’s a vector quantity, typically denoted by the symbol g, and is measured in units of acceleration (m/s²).
• Mathematically, gravitational field strength can be described by:
• [math] g = G * (M / r²) [/math]
• Where:
  – g is the gravitational field strength (m/s²)
  – G is the gravitational constant (6.67408e-11 N*m²/kg²)
  – M is the mass of the object (kg)
  – r is the distance from the center of the object (m)
• Also, can measure gravitational field by using the newton’s second law
• Which
• [math]\vec{F} = m \vec{a} \\
  \text{By gravitation, } \vec{F} = m \vec{g} \\
  \vec{g} = \frac{\vec{F}}{m} [/math]
• Gravitational field strength depends on:
  1. Mass of the object (more massive objects produce stronger fields)
  2. Distance from the object (field strength decreases with increasing distance)
  3. Distribution of mass (shape and orientation of the object)
• Gravitational field strength has significant implications in:
  1. Gravity and motion
  2. Planetary orbits and trajectories
  3. Tides and gravitational waves
  4. Gravitational lensing and bending of light
  5. Cosmology and the expansion of the universe
• Some examples of gravitational field strengths include:
  – Earth’s surface: approximately 9.8 m/s²
  – Moon’s surface: approximately 1.62 m/s²
  – Sun’s surface: approximately 28 times that of Earth’s surface

10. Newton’s third law of motion:

Newton’s Third Law, also known as the Law of Action and Reaction, states:
“For every action, there is an equal and opposite reaction.”

Figure 21 Action and reaction equal but in opposite direction

Figure 21 Action and reaction equal but in opposite direction

• This law applies to all interactions between objects, and it’s a fundamental principle in understanding how forces work.
• Some key aspects of Newton’s Third Law:
  – Forces always come in pairs (action-reaction pairs).
  – The forces are equal in magnitude and opposite in direction.
  – The law applies to all types of forces (friction, gravity, normal force, etc.).

Examples:
(1)

Figure 22 Two persons apply equal forces but in opposite direction

(2)

• Two balloons have been charged positively. They each experience a repulsive force from the other.
• These forces are of the same size, so each balloon (if of the same mass) is lifted through the same angle.

Figure 23 Two balloons have been same charged (positively)