# Relationship Between Torque and Speed: Detailed Explanations and Problem

The article discusses the relationship between torque and speed of the rotating body and its solved problems.

The torque and speed characterize the rotational motion. The angular speed is the rotation rate, while the torque is the force designed for rotational motion. The power connects the torque with speed by explaining how much energy is shared when the body revolves due to applied force.

Every rotating body has a specified output power capacity, whereas its speed and torque change.

The output power is set by the product of applied force and the linear distance it traveled per unit of time. Mathematically,

P = Fd/t ………………(*)

When force F is applied on the body at a certain distance r from its axis of rotation, the torque that acts on it is given by,

𝜏 = r * F

F = 𝜏/r …………..(1)

The relation between linear distance d and angular distance θ  is,

d=r *θ………………..(2)

Substituting (1) and (2) into (*),

P = (𝜏/r )/ (r*θ) t

Simplifying terms,

P = 𝜏θ/t………………(3)

The term θ/t  is the angular speed ω  of the body. i.e, ω = θ/t

P = 𝜏ω  ………………….(4)

So, the torque and speed are related in terms of power as,

𝜏=P/ω……………………(5)

The torque is inversely proportional to the speed and directly proportional to the power

## If the body has the power of 50 W rotating at a speed of 10 rad/s, what is the torque acting on it?

Given:

P = 50 W

To Find:

𝜏 = ?

Formula:

𝜏 = p/ω

Solution:

The torque acting on the body is calculated as,

𝜏 = P/ ω

Substituting all values,

𝜏 = 50/10

𝜏 = 5

The torque acting on the body is 5 Nm.

## The torque of 50 Nm acting on the car engine has the power of 150 W. Calculate how much it rotates in 10 sec?

Given:

𝜏 = 50 Nm

P = 150 W

t = 10 s

To Find: θ = ?

Formula:

𝜏 = P/ ω

Solution:

The angular distance travelled by the car’s wheels is calculated as,

𝜏 = P/ ω

ω = θ/t

𝜏 = P/ θ

Rearranging,

θ = P/ t𝜏

Substituting all values,

θ = 150/50 * 10

θ = 1500/50

θ = 30

The angular distance travelled by the car’s wheel is 30 rad/s.

## Torque and Speed relation in DC Motor

The torque and speed are inversely related in DC Motor.

Like motor devices convert electrical energy into mechanical energy, the DC motor also involves the electrical energy to rotational energy conversion. A specific voltage is supplied to the motor, which yields torque on the output shaft so that the motor begins rotating with the angular speed.

The torque, speed, and power are essential parameters that display the performance of the DC motor that involves energy conversion. The DC motor’s speed is defined by the input voltage needed to induce the torque on its shaft.

As you see, a vehicle requires less torque to move at a higher speed on a straight road. But the exact vehicle demands a high amount of torque while driving on the inclined road. That’s when its speed falls, but its power is constant.

There is two torque acting on the case of DC motor; one is load torque, and the other is induced torque. The load torque is the mechanical load employed on the shaft, while induced torque is developed by input current to drive the load torque at a particular speed.

During the driving onto the inclined road, the load torque on the shaft becomes larger than the induced torque. Hence, the motor’s speed becomes slower on such a road. That is why the torque and angular speed are inversely related in the DC motor

## Torque from Rotational Speed

There is a torque-speed graph for every DC motor, and its slope illustrates its characteristics.

The slope coinciding at a point on the y-axis is termed ‘stall torque’ 𝜏s ; which reveals maximum torque at there is no angular speed. Similarly, the point on the x-axis where the slope coincides is termed ‘no load speed’ ωn ; which reveals maximum speed as no torque is applied.

The linear curve connects the two maximum points in the graph, giving rise to the equations of torque and speed in DC motor as,

𝜏 = 𝜏s – ω/ 𝜏s ωn………….6

ω = (𝜏s – 𝜏)ωn/ 𝜏s……………..7

The rectangle area can be drawn below the torque-speed curve with one corner at the graph’s origin and the other is coinciding with the curve is depict the power of the DC motor

Since torque and speed have an inverse relationship, its power should be maximum at the point where ω = 0.5 ωn   and 𝜏 = 0.5/ 𝜏s .

The fixed output power of the DC motor can be estimated by substituting (6) and (7) into (4).

The power of a DC motor is also measured in Horsepower, showing how much energy the motor’s engine can yield per unit of time. As per equations (4) and (8), we comprehend that power is directly related to torque and speed. So the more horsepower the engine has, the quicker the motor can work

## The DC motor drives at 30 rad/s when no torque is acting on its shaft. Its speed is lowered to 20 rad/s when the input current induces torque. The motor stops driving when induced torque reaches its maximum value of 10 Nm. What is the original value of torque inducing on the DC motor?

Given:

𝜏s = 10 Nm

To Find:

𝜏 = ?

Formula:

𝜏 = 𝜏s – ω/𝜏sωn

Solution:

The torque induced by input current on dc motor is calculated as,

𝜏 = 𝜏s – ω/𝜏sωn

Substituting all values,

𝜏 = 10 – 20/10*30

𝜏 = 10 – 6.66

𝜏 = 3.4

The torque acting dc motor is 3.4 Nm.

## The DC motor moves at 50 rad/s when the torque of 20 Nm is induced. Calculate the DC motor’s horsepower when it stops moving at a torque of 30N.

Given

𝜏 = 20 Nm

𝜏s = 30 Nm

To Find:  P =?

Formula:

P = 𝜏s2 – 𝜏s𝜏 – ω𝜏s + ω𝜏

Solution:

The power of DC motor is calculated as,

P = 𝜏s2 – 𝜏s 𝜏- ω𝜏s + ω𝜏

Substituting all values,

P = 302 – 30 * 20 -50 * 20 – 50 * 30

P = 900 – 60 – 100 + 150

P = 890W

The horsepower of DC motor is calculated as,

1 HP = 745.7W.

So, P = 890/745.7 HP

P= 1.19HP

The power of the DC motor is 1.19HP.