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How to Understand Momentum, Work Done, Energy, Torque and Power

Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.

how-to-understand-momentum-work-done-energy-torque-and-power

What Is Momentum?

Momentum is the product of mass and velocity of a body.

If m is the mass of a body and v is its velocity, then:

momentum = mv

So the heavier an object is, the greater it's momentum when it's moving if its velocity doesn't change. If it moves faster, its momentum also increases even its mass doesn't change.

In a collision between two or more bodies, momentum is always conserved. This means that the total momentum of the bodies before the collision equals the total momentum of the bodies after the collision.

So if m1 and m2 are two bodies with velocities of u1 and u2 respectively before the collision and velocities of v1 and v2 after the collision, then:

m1u1 + m2u2 = m1v1 + m2v2

Elastic and inelastic collisions

Elastic collisions

In the case of a fully elastic collision (or almost totally elastic such as when two rigid snooker balls collide) kinetic energy is also conserved and not transformed into another form of energy such as heat. If an elastic body collides with a second stationary body, the momentum of the moving body is transferred to the stationary body and the first body becomes stationary.

A perfectly elastic collision of equal masses

A perfectly elastic collision of equal masses

Inelastic collisions

In this case, kinetic energy is not conserved and some of it is converted into heat, light or sound and deformation of the body. An example of a partially inelastic collision is when two pieces of modeling clay collide.

A perfectly inelastic collision between two bodies.

A perfectly inelastic collision between two bodies.

Example

Two bodies with mass 5 kg and 2 kg and velocities 6 m/s and 3 m/s respectively collide. After the collision the bodies remain joined. Find the velocity of the combined mass.

This is an example of a perfectly inelastic collision.

Let m1 = 5 kg

Let m2 = 2 kg

Let u1 = 6 m/s

Let u2 = 3 m/s

m1u1 + m2u2 = m1v1 + m2v2

Since the bodies are combined after the collision, v1 = v2. Let's call this velocity v.

So:

m1u1 + m2u2 = m1v1 + m2v2 = m1v+ m2v = (m1 + m2)v

Substituting:

(5)(6) + (2)(3) = (5 + 2)v

30 + 6 = 7v

So v = 36/7

What Is Kinetic Energy?

Kinetic energy is a related concept to that of momentum. A body has kinetic energy when it is in motion.

If a body of mass m is moving at a velocity v, then:

Kinetic energy = (1/2)mv2

The kinetic energy of a body at velocity v is the work that must be done on the body to accelerate it to that velocity.

Example

A rifle bullet of mass 4 grams is moving at a velocity of 1200 m/s.
Calculate its energy.

First convert to SI units, so 4 g = 0.004 kg

Kinetic energy = (1/2)mv2 = (1/2)0.004(1200)2 = 2880 joules

To put this into perspective, this is approximately the same impact energy that a 34kg (75 pound) weight would produce if dropped from chimney height off a two storey house. The bullet has a small mass, but the squared term in the equation increases the energy massively. Every time velocity is doubled, energy increases fourfold. If velocity is tripled, energy increases ninefold. An asteroid impacting on Earth can create so much destruction, not necessarily because of its mass, but because of its extremely high velocity that can be tens of thousands of kilometres per hour.

Notice that the amount of energy depends on the square of the velocity. So if velocity is doubled, energy becomes four times as much. If velocity is tripled, energy becomes nine times as great.

Kinetic energy.

Kinetic energy.

What Is Work?

The definition of work in physics is that "work is done when a force moves a body through a distance". If there is no movement of the point of application of a force, no work is done. So for instance, a crane that is simply holding a load at the end of its steel rope is not doing work. Once it starts hoisting the load, it is then doing work. When work is done there is energy transfer. In the crane example, mechanical energy is transferred from the crane to the load, which gains potential energy because of its height above the ground.

The unit of work is the joule.

If work done is W

distance is s

and the force applied is F

then

W = FsCos Θ where Θ is the angle between the force and displacement

Example

A force F = 50 N is applied to a box of mass 4 kg resting on the ground. Friction between ground and box results in a force opposing motion which is Fₛ = 2 N. Calculate the acceleration and work done sliding the box 3 m.

Start by writing the force equation. Use Newton’s Second Law. The sum of forces produces a net acceleration.

We can use a convention for writing the force equation. So consider the forces acting “forwards” and accelerating the box as positive and forces acting to decelerate the box as negative, Since the friction force Fₛ acts opposite to the direction of motion, its sign is negative, so Newton's second law says that the sum of forces equals the mass multiplied by the acceleration of a body and:

F + (- Fₛ ) = ma

Substituting for F and Fₛ in the first equation gives:

50 + (- 2) = 50 - 2 = 4 x a

Rearranging:

a = (50–2) / 4 = 12 m/s²

Work done on the box is the applied force multiplied by the distance. Work is done opposing friction plus accelerating the box. The net force accelerating force is 50 - 2 = 48 N. The displacement is in the direction of the force, so Θ = 0 and Cos Θ = 1, so:

Work done = FCos Θs = 50 x 1 x 3 = 150 joules

Note however that although the work done is due to the 50 N force, the net force that accelerates the box is 50 - 2 = 48 N.

What Are Moments, Couples and Torque?

When a force acts on an object, it produces what is known as a turning moment or just simply a moment. An example is when you push on the trunk of a small tree. This produces a turning moment about the base of the tree which is balanced by the tension in the trunk and the restraining force of the roots. If you push too hard, you exceed a breaking limit and the trunk snaps or the tree gets uprooted. The moment of a force about a point is the magnitude of the force multiplied by the perpendicular distance between the force and the point. When 2 forces act in opposite directions, this is known as a couple and the magnitude of the twisting force or couple is called torque. If the forces are both of magnitude F, and the perpendicular distance between them is d, then:

Torque T = Fd

As you can see, if the force is increased or the distance is increased, the torque becomes greater. So this is why it is easier to turn something if it has a larger diameter handle or knob. A tool such as a socket wrench with a longer handle has more torque.

Moment of a force.

Moment of a force.

Two forces constitute a couple. The magnitude is the torque

Two forces constitute a couple. The magnitude is the torque

In the diagram below, a cantilevered shelf has a force F applied at the end. This creates a clockwise turning moment Fd. At the wall, the force F is balanced by a force F (not shown) acting upwards and also a counter-clockwise reactive turning moment R = Fd, provided by the brick structure.

Turning moment of a cantilevered shelf and reaction moment at the wall.

Turning moment of a cantilevered shelf and reaction moment at the wall.

What Is a Gearbox Used For?

A gearbox is a device that converts high-speed low torque to lower speed and higher torque (or vice versa). Gearboxes are used in vehicles to provide the initial high torque required to get a vehicle moving and accelerate it. Without a gearbox, a much higher-powered engine with a resulting higher torque would be needed. Once the vehicle has reached cruising speed, lower torque is required (just sufficient to create the force required to overcome the force of drag and rolling friction at the road surface).

Gearboxes are used in a variety of other applications, including power drills, cement mixers (low speed and high torque to turn the drum), food processors and windmills (converting low blade speed to high rotational speed in the generator)

A common misconception is that torque is equivalent to power and more torque equals more power. Remember, however, torque is a turning force, and a gearbox that produces higher torque also reduces speed proportionately. So the power output from a gearbox is equal to the power in (actually a little less because of friction losses, mechanical energy being wasted as heat)

This gate valve has a large diameter turning handle to increase torque and make turning of the valve stem easier

This gate valve has a large diameter turning handle to increase torque and make turning of the valve stem easier

Measurement of Angles in Degrees and Radians

Angles are measured in degrees, but sometimes to make the mathematics simpler and elegant it's better to use radians which is another way of denoting an angle. A radian is the angle subtended by an arc of length equal to the radius of the circle. Basically "subtended" is a fancy way of saying that if you draw a line from both ends of the arc to the centre of the circle, this produces an angle with magnitude of 1 radian.

An arc length r corresponds to an angle of 1 radian

So if the circumference of a circle is 2πr = 2π (r) the angle for a full circle is 2π

And 360 degrees = 2π radians

1 radian is the angle subtended by an arc of length equal to the radius r

1 radian is the angle subtended by an arc of length equal to the radius r

Angular Velocity

Angular velocity is the speed of rotation of an object. Angular velocity in the "real world" is normally quoted in revolutions per minute (RPM), but it's easier to work with radians and angular velocity in radians per second so that the mathematical equations turn out simpler and more elegant. Angular velocity denoted by the Greek letter ω is the angle in radians that an object rotates through per second.

Angular velocity denoted by the Greek letter omega, is the angle in radians turned through per second

Angular velocity denoted by the Greek letter omega, is the angle in radians turned through per second

What Is the Relationship Between Angular Velocity, Torque and Power?

If the angular velocity is ω

and torque is T

Then

Power = ωT

Example

A shaft from an engine drives a generator at 1000 RPM
The torque produced by the shaft is 1000 Nm

How much mechanical power does the shaft produce at the input to the generator?

1 RPM corresponds to a speed of 1/ 60 RPS (revs per second)
Each revolution corresponds to an angle of 2π radians
So 1 RPM = 2π/60 radians per second
And 1000 RPM = 1000 (2π/60) radians per second

So ω = 1000 (2π/60) = 200π/6 radians per second

Torque T = 1000 Nm

So power = ωT = 200π/6 x 1000 = 104.72 kW

References

Hannah, J. and Hillerr, M. J., (1971) Applied Mechanics (First metric ed. 1971) Pitman Books Ltd., London, England.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2022 Eugene Brennan