(i) Seat belts should be worn in cars.
(ii) Head rests are necessary in a car to prevent neck injuries when there is a collision from the rear.
Driving forces and resistances to the motion of vehicles In problems about such things as cycles, cars and trains, all the forces acting along the line of motion will usually be reduced to two or three: the driving force forwards, the resistance to motion (air resistance, etc.) and possibly a braking force backwards.
Resistances due to air or water always act in a direction opposite to the velocity of a vehicle or boat and are usually more significant for fast-moving objects.
Tension and thrust
The lines joining the crate of supplies to the parachute described at the beginning of this chapter are in tension. They pull upwards on the crate and downwards on the parachute. You are familiar with tensions in ropes and strings, but rigid objects can also be in tension.
When you hold the ends of a pencil, one with each hand, and pull your hands apart, you are pulling on the pencil. What is the pencil doing to each of your hands? Draw the forces acting on your hands and on the pencil.
Now draw the forces acting on your hands and on the pencil when you push the pencil inwards.
Your first diagram might look like figure 3.18. The pencil is in tension so there is an inward tension force on each hand.
Force and motion
M1
3
When you push the pencil inwards the forces on your hands are outwards as in figure 3.19. The pencil is said to be in compression and the outward force on each hand is called a thrust.
If each hand applies a force of 2 units on the pencil, the tension or thrust acting on each hand is also 2 units because each hand is in equilibrium.
●
? Which of the above diagrams is still possible if the pencil is replaced by a piece of string?Resultant forces and equilibrium
You have already met the idea that a single force can have the same effect as several forces acting together. Imagine that several people are pushing a car. A single rope pulled by another car can have the same effect. The force of the rope is equivalent to the resultant of the forces of the people pushing the car. When there is no resultant force, the forces are in equilibrium and there is no change in motion.
EXAMPLE 3.4 A car is using a tow bar to pull a trailer along a straight, level road. There are resisting forces R acting on the car and S acting on the trailer. The driving force of the car is D and its braking force is B.
Draw diagrams showing the horizontal forces acting on the car and the trailer
(i) when the car is moving at constant speed
(ii) when the speed of the car is increasing
(iii) when the car brakes and slows down rapidly.
In each case write down the resultant force acting on the car and on the trailer.
SOLUTION
(i) When the car moves at constant speed, the forces are as shown in figure 3.20
forces on pencil tension Figure 3.18 forces on pencil thrust Figure 3.19 The forces on your hands
are inwards. The pencil is in tension.
The forces on your hands are outwards. The pencil is
Forces and Newton’
s laws of motion
52
M1
3
There is no resultant force on either the car or the trailer when the speed is constant; the forces on each are in equilibrium.
For the trailer: T − S = 0 For the car: D − R − T = 0
(ii) When the car speeds up, the same diagram will do, but now the magnitudes of the forces are different. There is a resultant forward force on both the car and the trailer.
For the trailer: resultant = T – S For the car: resultant = D – R – T
(iii) When the car brakes a resultant backward force is required to slow down the trailer. When the resistance S is not sufficiently large to do this, a thrust in the tow bar comes into play as shown in the figure 3.21.
For the trailer: resultant = T + S For the car: resultant = B + R − T Newton’s second law
Newton’s second law gives us more information about the relationship between the magnitude of the resultant force and the change in motion. Newton said that
●
● The change in motion is proportional to the force.
For objects with constant mass, this can be interpreted as the force is proportional to the acceleration.
Resultant force = a constant × acceleration
1Forces on trailer Forces on car
S R D
T T
Figure 3.20 Car travelling at constant speed tension in the towbar
Forces on trailer Forces on car
S R B
T T
Figure 3.21 Car braking
Force and motion
M1
3
The constant in this equation is proportional to the mass of the object: a more massive object needs a larger force to produce the same acceleration. For example, you and your friends would be able to give a car a greater acceleration than you would be able to give a lorry.
Newton’s second law is so important that a special unit of force, the newton (N), has been defined so that the constant in equation is actually equal to the mass. A force of 1 newton will give a mass of 1 kilogram an acceleration of 1 m s–2. The equation then becomes:
Resultant force = mass × acceleration
This is written: F = ma
The resultant force and the acceleration are always in the same direction. Relating mass and weight
The mass of an object is related to the amount of matter in the object. It is a scalar. The weight of an object is a force. It has magnitude and direction and so is a vector.
The mass of an astronaut on the moon is the same as his mass on the earth but his weight is only about one-sixth of his weight on the earth. This is why he can bounce around more easily on the moon. The gravitational force on the moon is less because the mass of the moon is less than that of the earth.
Forces and Newton’
s laws of motion
54
M1
3
When Buzz Aldrin made the first landing on the moon in 1969 with Neil Armstrong, one of the first things he did was to drop a feather and a hammer to demonstrate that they fell at the same rate. Their accelerations due to the gravitational force of the moon were equal, even though they had very different masses. The same is true on earth. If other forces were negligible all objects would fall with an acceleration g.
When the weight is the only force acting on an object, Newton’s second law means that
Weight in newtons = mass in kg × g in m s−2. Using standard letters:
W = mg
Even when there are other forces acting, the weight can still be written as mg. A good way to visualise a force of 1 N is to think of the weight of an apple. 1 kg of apples weighs approximately (1 × 10) N = 10 N. There are about 10 small to medium-sized apples in 1 kg, so each apple weighs about 1 N.
Note
Anyone who says 1 kg of apples weighs 1 kg is not strictly correct. The terms weight and mass are often confused in everyday language but it is very important for your study of mechanics that you should understand the difference.
EXAMPLE 3.5 What is the weight of
(i) a baby of mass 3 kg
(ii) a golf ball of mass 46 g?
SOLUTION
(i) The baby’s weight is 3 × 10 = 30 N. (ii) Mass of golf ball = 46 g
= 0.046 kg Weight = 0.046 × 10
Pulleys
M1
3
EXERCISE 3C Data: On the earth g = 10 m s–2. On the moon g = 1.6 m s–2. 1000 newtons (N) = 1 kilonewton (kN).
1 Calculate the magnitude of the force of gravity on the following objects on