 |
TORQUE AND HORSEPOWER |
See also ...
( Home → Science → Performance → Torque )
|
Torque and horsepower are the usual numbers quoted by car manufacturers to indicate how strong or powerful their engines are, under the assumption that we all know what they're talking about. Read this article to find out what these quantities mean. They're not as mysterious as you might think. |
The Concept of Torque
Torque and horsepower are two of the quantities that motor manufacturers are very keen on quoting for their engines. Usually they quote the maximum values that these quantities can attain, and the engine speed at which these peaks occur. For my SEAT Toledo with Superchip installed, the values are a maximum torque of 208 lb ft at about 2000 RPM, and a maximum power of about 135 BHP at around 4200 RPM (figures from Diesel Car magazine, April 2000).
To illustrate what we mean by torque and power, consider the diagram below:
In this diagram, a force is being applied to an axis, rather like a spanner being used to undo a tight nut. If the bar connecting the point of application of the force and the axis is rigid, then the point of application will tend to rotate around the axis. The ‘twisting’ force we are applying here is called a TORQUE. The magnitude of the torque is given by
In SI units, the units of torque are newton-metres (often written simply as Nm). Thus, 1 newton-metre is the torque arising from the application of a force of 1 newton (about the weight of a 100 g object) at a distance of 1 metre from the point of rotation. We can see from this equation that we can increase the magnitude of the torque by either applying more force F, or increasing the distance d between the point of application of the force and the axis. Thus, if a spanner cannot undo a tight nut, we either apply more force, or (more commonly) we use a longer spanner.
To visualise the magnitude of newton-metres, imagine holding a 1 kg bag of sugar at arms length. If your arm span is about 1 metre, then the torque exerted by the bag of sugar about your shoulder is about 10 Nm, since 1 kg weighs about 10 newtons.
The Concept of Power
To understand the concept of power, it is necessary first to go back to Newton’s second law, as considered in the previous section. Let us consider a constant force F, and assume that our object of mass M is initially at rest. The acceleration is F/M and the velocity at some time t after applying the force is given by:
The distance L travelled in reaching velocity v is equal to the average velocity over time t multiplied by the time t. Using the expression above and noting that the average velocity is equal to (v + v0)/2, we get
The quantity on the right hand side of this expression is the change in KINETIC ENERGY of the object in moving to velocity v, and is a measure of the change of energy of motion of the object. The product on the left hand side, FL, is defined as the WORK done on the object by the force F.
Now divide both sides of the above expression by the time t required to cover the distance L. Using (10), the result is:
The quantity on the left is the average POWER developed over the time t, and the quantity on the right is the change in kinetic energy per unit time. Generalising this expression, the power is defined by
Power = force ´ velocity
The meaning of the term ‘power’ can be expressed in the following terms:
the power developed is equal to the rate of change of kinetic energy
Power at Top Speed
At this point, an interesting observation can be made: if the car is at its maximum speed, then the kinetic energy is constant, and the total applied force F is zero, so according to the expressions above the power acting on the car is zero. However, it should not be forgotten that the applied force F is equal to the sum of all the applied forces. At the maximum speed, the tractive force supplied by the engine is equal to the sum of the aerodynamic and frictional forces. In terms of engine power,
engine power at max speed = P(drag) + P(friction)
where
P(Drag) is the power required to overcome aerodynamic drag
P(friction) is the power required to overcome other frictional forces.
So, even though the total power acting to change the kinetic energy of the car is zero, the power generated by the engine certainly is not.
Power in Terms of Angular Velocity
Getting back to applying torques about a point, suppose that, in the figure above, the point of application of the force is rotating at a rate of f rotations per second. Because the circumference of a circle is 2pd, where d is the radius, the velocity of the point of application is just 2pdf. For the case of rotational motion, the power is then given by:
Power = force ´ velocity = 2pdfF = 2p ´ Fd ´ f = 2p ´ torque ´ rotations per second
Where we have used the fact that
torque = force ´ distance = Fd
Symbolically, this is usually written as
,
where w is called the angular velocity, and is equal to the factor 2p ´ rotations per second.
To summarise, we have obtained two important results:
1. 1. TORQUE can be thought of as the application of a force about a rotational axis;
2. POWER is the product of torque and angular velocity (or force multiplied by velocity).
Units of Power and Torque
Finally, it is worth making a few comments about the units used to measure power and torque. In SI units, torque is measured in units of newton-metres, and power is measured in watts (joules/second). In the UK, torque is often measured in units of lb-ft. To convert between the two, we note that:
newton-metres = lb-ft ´ newtons per lb ´ metres per foot.
Since there are about 9.8 newtons in a kilogram, there are 9.8/2.204 newtons per pound. Similarly, there are 1/3.28 metres per foot. Therefore to convert from lb-ft to newton-metres, we use
newton-metres = lb-ft ´ 9.8/2.204 ´ 1/3.28 = lb-ft ´ 1.356
To convert from watts to horsepower (HP) we note that 1 HP = 746 Watts:
Power (HP) = Power (watts) / 746
To obtain the power in HP corresponding to a torque given in lb-ft and an engine speed in revs per minute (RPM), we note that, in our expression for torque, f = RPM/60, torque in newton-metres = torque in lb-ft ´ 1.356 and that Power in HP = power in watts / 746. Thus:
Power in HP = Torque in lb-ft ´ RPM ´ 2p ´ 1.356 / (60 ´ 746)
Or if we work out those numerical factors,
Power in HP = (Torque in lb-ft) ´ RPM / 5253.5
The factor of 2p arises from the need to use angular velocity (= 2pf) in this expression.
One curiosity of this last result is that the power in HP and torque in lb-ft are numerically equal at 5253 RPM, and if you look at any dynamometer plot you will see that the power and torque curves cross at this point. So, if your car develops 200 BHP at 5253 RPM, then the torque developed is equal to 200 lb-ft.