1.1 Physical Quantities & Units

Estimate the time it takes light to cross the nucleus of a hydrogen atom.

A foil made from gold has a thickness of 0.132 µm. A gold atom has a radius of 146 pm.

The atomic mass unit u is a common unit used in nuclear physics. 1u is equivalent to 931.5 MeV c^–2

Show that 1u is approximately equal to the mass of a proton in kg.

An acre is a unit of land measurement approximately equal to 4050 m² which is roughly equivalent to the size of a football field.

An Astronomical Unit, or 1 AU, is defined as the average distance between the Earth and the Sun and is approximately equal to 150 million km.

Fig. 1.1 shows two taut, light ropes that keep a pole vertically upright by applying two tension forces, one of magnitude 200 N and one of magnitude T. 

Fig. 1.1

By constructing a scale diagram, determine the weight of the pole W and the magnitude of T. 

A canoeist can paddle at a speed of 3.8 m s^–1 in still water. The canoeist encounters an opposing current, moving at a speed of 1.5 m s^–1 at 30° to their original direction of travel, as shown in Fig. 1.2. 

Fig. 1.2

Using Fig. 1.2 to construct a scale diagram, determine the magnitude of the canoeist's resultant velocity. 

The boat shown in Fig. 1.3 is being towed at a constant velocity by a towing rope, which exerts a tension force F_T = 2500 N.

There are two resistive forces indicated; the force of the water on the keel F_K and the force of the water on the rudder, F_R. 

Fig. 1.3

By scale diagram or otherwise, determine the magnitude of F_R. 

Another boat wishes to cross a river. The river flows from west to east at a constant velocity of 35 cm s^–1 and the boat leaves the south bank, due north, at 1.5 m s^–1 as shown in Fig. 1.4.

Fig. 1.4

Construct a scale diagram to show that the new direction of the boat is about 13° East of North.

Estimate the speed of an apple as it hits the ground after falling from a tree.

Estimate the number of kilowatt-hours used by a lightbulb if switched on for a whole day.

Estimate the density, in g cm⁻³, of the head of an adult person.

The radius of the Earth is approximately 6.4 × 10⁶ m.

Estimate the number of people it would take to circle the Earth holding hands.

Derive the SI base unit of force.

As a spherical object of radius moves through the air at speed , it experiences a force due to air resistance.

The resistive force is given by 

Where  is a constant.

Derive the SI base unit of the constant .

The relationship can be used to model a spherical raindrop falling vertically in the air.

Fig 1.1 shows the forces acting on the raindrop as it accelerates downwards.

The raindrop has a radius of 1.5 mm and falls at a velocity of 3.5 m s⁻¹. The constant k has a magnitude of 0.0124 in air.

Other than the resistive force F described in (b), two other forces, X and Y, act on the raindrop drop as it falls through the air. 

On a windy day, the resultant force on the raindrop is 105 µN at an angle of θ to the vertical as shown in Fig. 1.2. 

Use your answer to (c)iii to show that the angle θ between the resultant and the vertical is about 47°.

Two gusts of wind exert forces of magnitude 5.5 N and 11.9 N on a wind turbine at a point O. Both forces act away from point O and the angle between them is 30°.

Fig. 1.1

Fig 1.2 shows a turbine that is used to generate electrical power P from wind travelling with velocity v.

The useful power output P of the turbine is given by

Where L is the length of each blade, ρ is the density of air, v is the wind velocity and C_P is a constant.

Show that the constant C_P has no units.

A local council is looking into building wind turbines to supply power to 2500 households in a village. The average energy consumption of a household in the village is 3000 kWh a year.

Fig. 1.3. shows an image from a catalogue detailing the different sizes and power outputs of available wind turbines.

They decide to build 8 of the same wind turbine to supply the entire village with electricity. 

Fig. 1.1 shows a 2 tonne wrecking ball being used on a demolition site. 

The ball is suspended from a cable which makes an angle θ with the vertical. The ball is pulled into the position shown by a rope that is kept horizontal. The tension in the rope is 6400 N.

Write expressions for tension T using

Determine

On the same demolition site, there is a uniform beam supported by two light cables, AB and AC, which are attached to a single steel cable from a crane, as shown in Fig. 1.2.

The beam is stationary and in equilibrium.

Fig. 1.2

Draw the vector triangle for this situation labelling the tension in both cables and the weight of the beam. 

The tension in the cable AB is 9 N and the tension in the cable AC is 12 N.

Calculate the resultant force required in the beam BC to keep the system in equilibrium.