In this article, we will learn how to find angular acceleration of a wheel, and solve some problems related to the angular acceleration of a wheel.

**The angular acceleration is a change in the angular velocity caused due to variations in the angular movement of a wheel on experiencing a torque equivalent to the moment of inertia of a wheel and the force implied tangential to a wheel.**

**Angular Acceleration of a Wheel from Angular Velocity**

**The angular acceleration is simply the ratio of change in the angular velocity of a wheel on rotation with respect to time.** This is given by the equation as below:-

α = ω_{2} – ω_{1} / t_{2}-t_{1}

Where α is a angular acceleration,

ω_{1} and ω_{2} are final and initial angular velocities respectively and

t_{1} and t_{2} are final and initial time during which the change has occurred.

**Read more on How To Find Angular Acceleration From Angular Velocity: Problem And Examples.**

**Example 1: Calculate the angular acceleration of a wheel of a car in a motion knowing that the angular velocity of a car was 30rad/s, which increases to 80rad/s in 40 seconds.**

**Solution:** We have, ω_{2} =80rad/s

ω_{1}=30rad/s

T=40sec

Hence,

α = ω_{2} – ω_{1} / t_{2}-t_{1}

α = 80 – 30/ 40 = 1.25 rad/s^{2}

The angular acceleration of a wheel of a car is 1.25rad/s^{2}.

**Relation between Angular Acceleration and Tangential Acceleration**

We know that the tangential velocity of a body in an angular motion is related to the angular velocity by the following equation

v = ωr —(2)

**The tangential acceleration of an object is a change in the tangential velocity with respect to varying time.**

a = dv/dt

Substituting eqn(2) here

a=r dω/dt

Hence,

a=rα —(3)

Where α is the angular acceleration.

α =a/r —(4)

**The angular acceleration of a wheel is directly proportional to the tangential acceleration of a wheel and inversely proportional to the radius of a wheel. If the radius of a wheel increases then the variations seen in the θ will be minimized.**

**Read more on How To Find Tangential Acceleration: Problems And Examples.**

**Example 2: An object moving in a circular path of radius 12m accelerates at a rate of 4m/s**^{2}. Calculate the angular acceleration of an object.

^{2}. Calculate the angular acceleration of an object.

**Given:** a=4m/s^{2},

r=12m

We have,

α=a/r=4/12 =0.33rad/s

The angular acceleration of an object is 0.33rad/s.

**Acceleration due to Torque**

A force has to be applied to the wheel to set it in motion. The force applied has to be tangential to the wheel to set a wheel in the translational motion.

**Torque on a wheel is determined by the force applied and the length of the displacement of a wheel due to force. Torque on a wheel reply upon how much is the mass of a wheel. The greater the mass, the more force will be required, and hence torque produced will be of greater quantity.**

The torque experience on a wheel is given by

𝜏 =F* displacement

𝜏 =Ma* R

Substituting eqn(3) in the above equation we get

𝜏 =MR^{2}α

**The tendency of a body to resist the angular acceleration due to its mass is called the moment of inertial and is the product of the entire mass of the object and the square of its distance from the axis of rotation.**

I=MR^{2}

Therefore,

𝜏 =I α —(5)

Hence,

α =𝜏/I —(6)

**Angular acceleration**** is a ratio of torque experienced on a body to its moment of inertia.**

**Read more on How To Find Tension To Torque.**

**How to Calculate Moment of Inertia of a Wheel?**

Less the moment of inertia, less will be the torque on the body.

**Suppose a wheel has a mass ‘M’ and a radius of a wheel be ‘R’, then the moment of inertia of a wheel is equal to the total moment of inertia due to rim and all spokes.**

In the below diagram there are 8 spokes of a wheel and the mass of each spoke is one-third the mass of a wheel.

I=I_{rim}+I_{spoke}

=MR^{2}+8 ( 1/3 MR^{2} )

Length of a spoke is equal to the radius of a wheel, hence,

=MR^{2}+8 (1/3 MR^{2})

=1+1/3 MR^{2}

=11/3 MR^{2} —(7)

Hence, we now have a moment of inertia of a wheel.

Substituting Eqn(7) in Eqn(5), the angular acceleration of a wheel is

α =3𝜏/11 MR^{2}

**Example 3: Consider a wheel of radius 20cms and mass 2kg. A force of 20N is applied on a wheel and the wheel travels a distance of 20meters. Then calculate the angular acceleration of a wheel.**

**Given:** r=20cm,

m=2kg,

x=20m,

F=20N

Torque on a wheel is

𝜏 =F*x=20N* 20m=400Nm

Hence, the angular acceleration of a wheel is

α=3𝜏/ 11MR^{2}

=3*400N/11*2kg*(0.2m)^{2}

=1.36rad/s^{2}

The angular acceleration of a wheel is 1.36rad/s2.

**Read more on Angular Acceleration.**

**Frequently Asked Questions**

**Q1. Consider a giant wheel of mass 5kg and radius 120cm rotating at a speed of 2 revolutions per second accelerates and attains the angular velocity of 5rpm in 25 seconds. Calculate the tangential acceleration of a wheel and torque experienced tangential to a wheel.**

**Given:** M=5kg, r=120cm=1.2m

ω_{i}=2rpm=2*2π/60=π/15rad/s

ωf=5rpm=5 * 2π * 60 =π/6 rad/s

t=25 seconds

The angular acceleration of a wheel is

α = ω_{f}– ω_{i} / t_{2}-t_{1}

=π/6-π/15*25=π/ 250 rad/s^{2}

The tangential acceleration of a wheel is

a=rα

=1.2m*π/250 rad/s^{2}=0.015m/s^{2}

Moment of Inertia of a wheel is

I=11/3 MR^{2}

=11/3 * 5* (1.2)^{2}=26.4 kg.m^{2}

Hence, the torque on a wheel is

α =𝜏/I

𝜏 =α I

π rad/s^{2}*26.4kgm^{2}=0.33Nm

**Q2. If there are 12 number of spoke on a wheel, then calculate the moment of inertia of a wheel.**

The moment of inertia due to spokes will be

I_{spoke}=12 ( 1/3 MR^{2})=4MR^{2}

The moment of inertial of a rim is

I_{rim}=MR^{2}

Therefore, the moment of inertia of a wheel is

I =I_{rim} + I_{spoke}=4MR^{2}+ MR^{2}=5MR^{2}

**Briefly describe the motion of a wheel.**

A wheel shows a rotational and translational motion. It will show only the rotational motion when connected with an axle.

**One rotation of a wheel is equal to the angle 4****π****. In translational motion, the length traveled by the wheel in one revolution on rotation is equal to the circumference of a wheel.**

**If the radius of a wheel increases 10 times then what will be the effect on the tangential and the angular acceleration of a wheel?**

The angular acceleration and tangential acceleration is related to the radius of a wheel by the equation a=r**α**

**If the radius of a wheel increases 10 times, then the tangential acceleration will increase by 10 times, while the angular acceleration will remain unaffected but the torque required to displace the angle θ will be more.**