An inclined plane makes an angle of 60° with horizontal. A disc at rest starts rolling down this inclined plane without slipping. Its linear speed when it lowers by a height of 3 m, is?

To find the linear speed of the disc as it rolls down an inclined plane, we can use the principles of energy conservation. When the disc rolls down the incline, its potential energy is converted into kinetic energy. Let's break this down step by step.

Understanding the Energy Transformation

Initially, the disc has gravitational potential energy due to its height above the ground. As it rolls down the incline, this potential energy is converted into two forms of kinetic energy: translational kinetic energy (due to its linear motion) and rotational kinetic energy (due to its spinning). The total mechanical energy remains constant if we neglect friction and air resistance.

Formulas Involved

We can express the potential energy (PE) and kinetic energy (KE) as follows:

• Potential Energy (PE): PE = mgh
• Translational Kinetic Energy (KE_trans): KE_trans = (1/2)mv²
• Rotational Kinetic Energy (KE_rot): KE_rot = (1/2)Iω²

Here, m is the mass of the disc, g is the acceleration due to gravity (approximately 9.81 m/s²), h is the height, v is the linear speed, I is the moment of inertia, and ω is the angular velocity.

Calculating the Height

Given that the disc lowers by a height of 3 m, we can calculate the potential energy at this height:

PE = mgh = mg(3)

Moment of Inertia for a Disc

The moment of inertia I for a solid disc about its center is given by:

I = (1/2)mr²

For rolling without slipping, the relationship between linear speed v and angular speed ω is:

ω = v/r

Combining Energies

The total kinetic energy when the disc is rolling down the incline can be expressed as:

KE_total = KE_trans + KE_rot = (1/2)mv² + (1/2)Iω²

Substituting the moment of inertia and the relationship between v and ω gives:

KE_total = (1/2)mv² + (1/2)(1/2)mr²(v/r)²

After simplifying, we find:

KE_total = (1/2)mv² + (1/4)mv² = (3/4)mv²

Setting Up the Energy Equation

Now, we can set the initial potential energy equal to the total kinetic energy at the bottom of the incline:

mgh = (3/4)mv²

Notice that the mass m cancels out:

gh = (3/4)v²

Solving for Linear Speed

Now, substituting g = 9.81 m/s² and h = 3 m:

9.81 * 3 = (3/4)v²

29.43 = (3/4)v²

Multiplying both sides by 4/3 gives:

v² = (4/3) * 29.43

v² = 39.24

Taking the square root:

v ≈ 6.26 m/s

Final Result

Thus, the linear speed of the disc when it has lowered by a height of 3 m is approximately 6.26 m/s.