Two wheels having radii in the ratio 1:3 are connected by a common belt. If the smaller wheel is accelerated from rest at a rate $of 1.5 rad/s^2$  for 10 s. Find the angular velocity of bigger wheel. 

Two wheels with radii in the ratio of 1:3 are connected by a common belt. The smaller wheel is accelerated from rest at a rate of 1.5 rad/s² for 10 seconds. We are asked to find the angular velocity of the bigger wheel.

Step-by-Step Solution

Step 1: Understand the Given Data

• The ratio of the radii of the two wheels is 1:3.
• Let the radius of the smaller wheel be r₁ = x and the radius of the bigger wheel be r₂ = 3x.
• The angular acceleration of the smaller wheel is α₁ = 1.5 rad/s².
• The time during which the smaller wheel is accelerated is t = 10 s.

Step 2: Calculate the Angular Velocity of the Smaller Wheel

To calculate the angular velocity of the smaller wheel, we use the formula for angular velocity:

ω₁ = ω₀ + α₁ × t    

Since the smaller wheel starts from rest, ω₀ = 0. Substituting the values:

ω₁ = 0 + (1.5 rad/s²) × (10 s)        ω₁ = 15 rad/s    

The angular velocity of the smaller wheel after 10 seconds is ω₁ = 15 rad/s.

Step 3: Relate the Linear Velocities of the Two Wheels

Since the two wheels are connected by a common belt, the linear velocities at the edges of both wheels must be equal:

v₁ = v₂    

The linear velocity can be expressed in terms of the angular velocity and radius:

v₁ = ω₁ × r₁ and v₂ = ω₂ × r₂    

Equating the linear velocities:

ω₁ × r₁ = ω₂ × r₂    

Step 4: Substitute the Known Values

We know that:

r₁ = x and r₂ = 3x    

Substitute these values into the equation:

ω₁ × x = ω₂ × (3x)    

We can cancel out x from both sides (assuming x ≠ 0):

ω₁ = 3 × ω₂    

Step 5: Solve for the Angular Velocity of the Bigger Wheel

Now, substituting the value of ω₁:

15 = 3 × ω₂    

Dividing both sides by 3 gives:

ω₂ = 15 ÷ 3        ω₂ = 5 rad/s    

Final Answer

The angular velocity of the bigger wheel is ω₂ = 5 rad/s.