mechanics Flashcards
Scalars
A scalar is a quantity which only has a magnitude (size)
e.g. Distance is a scalar quantity because it describes how far an object has travelled overall, but not the direction it has travelled in.
vector
A vector is a quantity which has both a magnitude and a direction
e.g. Displacement is a vector quantity because it describes how far an object is from where it started and in what direction
Moment
A moment is the turning effect of a force
Moments occur when forces cause objects to rotate about some pivot
The moment of a force is given by
Moment (N m) = Force (N) × perpendicular distance from the pivot (m)
The Principle of Moments
For a system to be in equilibrium, the sum of clockwise moments about a point must be equal to the sum of the anticlockwise moments (about the same point)
Couples
A couple is a pair of equal and opposite coplanar forces that acts to produce rotation only
A couple consists of a pair of forces that are:
Equal in magnitude
Opposite in direction
Perpendicular to the distance between them
moment of a couple
Couples produce a resultant force of zero, so, due to Newton’s Second law (F = ma), the object does not accelerate
Unlike moments of a single force, the moment of a couple doesn’t depend on a pivot
The moment of a couple is equal to:
Force × Perpendicular distance between the lines of action of the forces
Centre of Mass
The centre of mass of an object is the point at which the weight of the object may be considered to act
- The position of the centre of mass of uniform regular solid is at its centre
- For example, for a person standing upright, their centre of mass is roughly in the middle of the body behind the navel, and for a sphere, it is at the centre
- For symmetrical objects with uniform density, the centre of mass is located at the point of symmetry
Moments in levers
In a lever, an effort force acts against a load force by means of a rigid object
rotating around a pivot. Levers are really useful in situations where you
need a larger turning effect. Examples include spanners, wheelbarrows and
scissors. They increase the distance from the pivot a force is applied, so
you need less force to get the same moment.
levers e.g
Centre of mass and moments
An object will topple over if the line of action of its weight (drawn down from the centre of mass) falls outside its base area. This is because a resultant moment occurs, which provides a turning force.
The higher the centre of mass, and the smaller the base area, the less stable the object will be. An object will be very stable if it has a low centre of mass and a wide base area.
Forces on supports
If an object is being held up by supports (e.g. chair legs, car tyres, etc.), the force acting on each support won’t always be the same. The closer the object’s centre of mass is to a support, the stronger the force on the support.
Centre of gravity v Centre of mass
- In a uniform gravitational field, the centre of gravity is identical to the centre of mass
- The centre of mass does not depend on the gravitational field
- Since weight = mass × acceleration due to gravity, the centre of gravity does depend on the gravitational field
- When an object is in space, its centre of gravity will be more towards the object with the larger gravitational field
Acceleration
Acceleration is the rate of change of velocity.
Displacement simple
Displacement = average velocity x time.
If acceleration is constant, the average velocity is just the average of the initial and final velocities, so:
Displacement advanced
velocity (suvat)
SUVAT Equations
The SUVAT equations are the equations of motion used for objects in constant acceleration
They contain the following variables:
s = displacement (m)
u = initial velocity (m s-1)
v = final velocity (m s-1)
a = acceleration (m s-2)
t = time (s)
Plotting displacement-time graphs
You need to be able to plot displacement- time graphs for moving objects. The suvat equations from the last topic can be used to work out values to plot. Displacement is plotted on the y-axis and time on the x-axis.
Acceleration on displacement-time graphs
The gradient of a displacement-time graph shows velocity. Acceleration is the rate of change of velocity, so on a distance-time graph, acceleration is the rate of change of the gradient.
A graph of displacement against time for an accelerating object always produces a curve. If the object’s accelerating at a uniform rate, then the rate of change of the gradient will be constant. Acceleration is shown by a curve with an increasing gradient (like the one in the example above). Deceleration is shown by a curve with a decreasing gradient.
velocity on displacement-time graphs
When velocity is constant, the displacement-time graph is a diagonal straight line. As you saw on the previous page, the gradient of a displacement-time graph shows velocity.
Instantaneous Speed / Velocity
The instantaneous speed (or velocity) is the speed (or velocity) of an object at any given point in time
This could be for an object moving at a constant velocity or accelerating
An object accelerating is shown by a curved line on a displacement – time graph
An accelerating object will have a changing velocity
To find the instantaneous velocity on a displacement-time graph:
Draw a tangent at the required time
Calculate the gradient of that tangent
Average Speed / Velocity
- The average speed (or velocity) is the total distance (or displacement) divided by the total time
- To find the average velocity on a displacement-time graph, divide the total displacement (on the y-axis) by the total time (on the x-axis)
- This method can be used for both a curved or a straight-line on a displacement-time graph
Uniform & Non-Uniform Acceleration Graphs
Velocity-Time Graph
- Slope equals acceleration
- The y-intercept equals the initial velocity
- A straight line represents uniform acceleration
- A positive slope represents an increase in velocity (acceleration) in the positive direction
- A negative slope represents an increase in velocity (acceleration) in the negative direction
- A curved line represents the non-uniform acceleration
- A horizontal line (zero slope) represents motion with constant velocity
- The area under the curve equals the displacement or distance travelled
Acceleration-Time Graph
The slope is meaningless
The y-intercept equals the initial acceleration
A horizontal line (zero slope) represents an object undergoing constant acceleration
The area under the curve equals the change in velocity
Motion of a Bouncing Ball
For a bouncing ball, the acceleration due to gravity is always in the same direction (in a uniform gravitational field such as the Earth’s surface)
Since the ball changes its direction when it reaches its highest and lowest point, the direction of the velocity will change at these points
At point A (the highest point):
* The ball is at its maximum displacement
* The ball momentarily has zero velocity
* The velocity changes from positive to negative as the ball changes direction
* The acceleration, g, is still constant and directed vertically downwards
At point B (the lowest point):
* The ball is at its minimum displacement (on the ground)
* Its velocity changes instantaneously from negative to positive, but its speed (magnitude) remains the same
* The change in direction causes a momentary acceleration (since acceleration = change in velocity / time)
Newton’s First Law
An object will remain at rest or move with constant velocity unless acted on by a resultant force.
This means that an object at rest / moving with a constant velocity will remain at rest / moving with a constant velocity unless acted upon by a resultant force
A resultant force is required to change the motion of an object in order to speed up, to slow down or to change direction
Newton’s Second Law
The resultant force acting on an object with a constant mass is directly proportional to its acceleration
- This acceleration always acts in the same direction as the resultant force
- When unbalanced forces act on an object, the object experiences a resultant force
- If the resultant force acts along the direction of the object’s motion, the object will Speed up (accelerate) or Slow down (decelerate).
- If the resultant force acts on an object at an angle to its direction of motion, it will Change direction
Newton’s Second Law and Momentum
The resultant force on an object is equal to its rate of change of momentum
This change in momentum is in the same direction as the resultant force
Newton’s Third Law
If Object A exerts a force on Object B, then Object B will exert a force on Object A which is equal in magnitude but opposite in direction
When two objects interact, the forces involved arise in pairs
These are often referred to as third-law pairs
A Newton’s third law force pair must be:
* The same type of force
* The same magnitude
* Opposite in direction
* Acting on different objects
Newton’s Third Law example
- The foot pushes on the ground and the ground pushes back on the foot
- Both of these forces are the normal contact force (sometimes called the support force or the normal reaction force)
- The forces are of equal magnitude
- The forces are opposite in direction
- The forces are acting on different objects (the foot and the ground)
What is freefall?
Freefall is when there’s gravity acting on an object and nothing else.
It’s defined as the motion of an object undergoing an acceleration of ‘g’.
You need to remember:
• Acceleration is a vector quantity — and ‘g’ acts vertically downwards.
• The only force acting on an object in freefall is its weight.
• Objects can have an initial velocity in any direction and still undergo freefall as long as the force providing the initial velocity is no longer acting.
Galileo’s freefall investigations
All objects in freefall accelerate to the ground at the same rate.
Galileo believed that all objects fall at the same rate. The problem in trying to prove it was that free-falling objects fell too quickly for him to be able to take any accurate measurement, and air resistance affects the rate at which objects fall. He measured the time a ball took to roll down a smooth groove in an inclined plane. Rolling the ball down a plane slowed down the ball’s fall as well as reducing the effect of air resistance.
By rolling the ball along different fractions of the total length of the slope, he found that the distance the ball travelled was proportional to the square of the time taken. The ball was accelerating at a constant rate.
Why do all objects fall at the same rate?
Newton’s 2nd law explains this nicely — consider two balls dropped at the same time, ball 1 being heavy and ball 2 being light.
Then use Newton’s 2nd law to find their acceleration:
Freefall and the equations of motion
there’s a sign convention: upwards is positive, downwards is negative.
• g is always downwards, so it’s usually negative.
• t is always positive.
• u and v can be either positive or negative.
• s can be either positive or negative.
Projectiles - Time of flight (total time):
Time of flight (total time): how long the projectile is in the air.
For typical projectile motion, the time to the maximum height is half of the total time
Projectiles - maximum height attained
Maximum height attained: the height at which the projectile is momentarily at rest
This is when the vertical velocity component = 0
When the projectile is released and lands on the ground the projectile is at its maximum height when half of its total time has elapsed
Projectiles - Range
Range: the horizontal distance travelled by the projectile
Projectiles - Range
Range: the horizontal distance travelled by the projectile
Example of suvat used to find height range and time of flight
Lift
Lift is an upwards force on an object moving through a fluid. It happens when the shape of an object causes the fluid flowing over it to change direction.
The force acts perpendicular to the direction in which the fluid is flowing.
Terminal speed
Terminal speed (or terminal velocity) happens when frictional forces equal the driving force. An object will reach a terminal speed at some point if there’s a driving force that stays the same all the time, and a frictional or drag force (or collection of forces) that increases with speed
Motion graphs for terminal velocity
Terminal speed in a fluid
Things falling through any fluid will reach a terminal speed. You can calculate the terminal velocity of a ball bearing (a little steel ball) in a viscous (thick) liquid by setting up an experiment like this:
• Put elastic bands around the tube of viscous liquid at fixed distances using a ruler, then drop a ball bearing into the tube, and use a stopwatch to record the time at which it reaches each band.
• Repeat this a few times to reduce the effect of random errors on your results, using a strong magnet to remove the ball bearing from the tube.
• Calculate the times taken by the ball bearing to travel between consecutive elastic bands and calculate an average for each reading.
Use the average times and the distance between bands to calculate the average velocity between each pair of elastic bands.
• You should find that the average velocity increases at first, then stays constant — this is the ball bearing’s terminal velocity in the viscous liquid used.
• You can then try the same experiment for different liquids and see how they compare.
Terminal velocity of a parachutist
When something’s falling through air, the weight of the object is a constant force accelerating the object downwards. Air resistance is a frictional force opposing this motion, which increases with speed
• So before a parachutist opens the parachute, exactly the same thing happens as with the car:
Elastic and inelastic collisions
An elastic collision is one where momentum is conserved and kinetic energy is conserved - i.e. no energy is dissipated as heat, sound, etc. Kinetic energy is the energy that an object has due to its motion.
If a collision is inelastic, it means that some of the kinetic energy is converted into other forms during the collision. Linear momentum is always conserved in inelastic collisions though.
The equation for kinetic energy is:
Impulse
When an external resultant force acts on an object for a very short time and changes the object’s motion, we call this impulse.
For example:
Kicking a ball
Catching a ball
A collision between two objects
Impulse is the product of the force applied and the time for which it acts
Measuring impulse
Because the force is acting for only a short time, it is very difficult to directly measure the magnitude of the force or the time for which it acts
Instead, it can be measured indirectly
Change in momentum is equal to impulse
Therefore, change in momentum can be used to measure impulse indirectly
These equations are only used when the force F is constant
Impulse, like force and momentum, is a vector quantity
The impulse is always in the direction of the resultant force
A small force acting over a long time has the same effect as a large force acting over a short time
Force time graph
Impulse is the area under a force-time graph — this is really handy for solving problems where the force changes.
Vehicle safety features shown on graph
The reduced force due to the increase in contact time can be shown on a force-time graph
For the same change in momentum, which depends on the mass and speed of a vehicle, the increase in contact time will result in a decrease in the maximum force exerted on the vehicle and passenger
This is demonstrated by a lower peak and wider base on a force-time graph
Forces at an angle
Sometimes the direction of movement of an object is different from the direction of the force acting on it.
If the force is at an angle θ to the object’s displacement, the work done is calculated by:
W = Fs cos θ
Where θ is the angle, in degrees, between the direction of the force and the motion of the object
When θ is 0 (the force is in the direction of motion) then cos θ = 1 and W = Fs
For horizontal motion, cos θ is used
For vertical motion, sin θ is used
Always consider the horizontal and vertical components of the force
The component needed is the one that is parallel to the displacement
Power
Power is the rate of doing work or the rate of energy transfer
Power is calculated by the equation:
The equation shows that the power is increased if:
There is a greater energy transfer (work done)
The energy is transferred (work is done) over a shorter period of time
Force-Displacement Graph
The work done by a force acting over a distance can also be found from a force-displacement graph
If the force is not constant and is plotted against the displacement of the object:
The work done is equal to the area under the force-displacement graph
This is because:
**Work done = Force × Displacement
**
The work done is therefore equivalent whether there is:
* A small force over a long displacement
* A large force over a small displacement
The principle of conservation of energy
Energy cannot be created or destroyed. Energy can be transferred from one form to another but the total amount of energy in a closed system will not change.
Efficiency
The ratio of the useful power output from a system to its total power input.
- If a system has high efficiency, this means most of the energy transferred is useful
- If a system has low efficiency, this means most of the energy transferred is wasted
- When electrical energy is converted to light in a lightbulb, the light energy is useful, and the heat energy produced is wasted
Kinetic energy
Kinetic energy is the energy of anything moving, which you work out from:
Gravitational potential energy
There are different types of potential energy, e.g. gravitational and elastic.
Gravitational potential energy is the energy something gains if you lift it up.
You work it out using:
Elastic potential energy
Elastic potential energy (elastic stored energy) is the energy stored in, say,
a stretched rubber band or spring. If you need, you can work this out using:
Types of Energy
