A solid sphere ,a hollow sphere and a disc,all having same mass and radius are placed on the top of an inclined plane and released .the friction coefficients between the objects and the incline are same and not sufficient to allow pure rolling .least time will be taken in reaching the bottom by

1) solid sphere

2)hollow sphere

3 disc

4)all will take same time

To determine which object will take the least time to reach the bottom of the inclined plane, we need to consider how each shape rolls down the incline under the influence of gravity, especially since they all have the same mass and radius, and the friction is not sufficient for pure rolling. This means that all objects will slide down the incline rather than roll without slipping.

Understanding the Motion of Each Object

When an object slides down an incline, its motion can be analyzed using the concepts of translational and rotational motion. However, since the friction is not enough for pure rolling, we can focus primarily on translational motion, which is influenced by the object's shape and mass distribution.

Acceleration on the Incline

The acceleration of an object sliding down an incline can be derived from Newton's second law. The net force acting on the object is the component of gravitational force acting down the slope, which can be expressed as:

• For a solid sphere: F = m * g * sin(θ)
• For a hollow sphere: F = m * g * sin(θ)
• For a disc: F = m * g * sin(θ)

Here, m is the mass, g is the acceleration due to gravity, and θ is the angle of the incline. The acceleration of each object down the incline can be expressed as:

• a = g * sin(θ) / (1 + k^2/r^2)

where k is the radius of gyration, which depends on the shape of the object:

• Solid sphere: k^2 = (2/5)r^2
• Hollow sphere: k^2 = (2/3)r^2
• Disc: k^2 = (1/2)r^2

Calculating Effective Acceleration

Now, substituting the values of k for each object, we can find the effective acceleration:

• Solid sphere: a = (5/7)g * sin(θ)
• Hollow sphere: a = (3/5)g * sin(θ)
• Disc: a = (2/3)g * sin(θ)

Time to Reach the Bottom

The time taken to reach the bottom of the incline can be calculated using the kinematic equation:

• s = ut + (1/2)at²

Since the initial velocity u is zero, we can simplify this to:

• s = (1/2)at²

Rearranging gives us:

• t = sqrt(2s/a)

Since all objects have the same mass and radius, and they start from the same height s, the time taken will be inversely proportional to the square root of their respective accelerations. Therefore, the object with the highest acceleration will take the least time to reach the bottom.

Comparing Accelerations

From our earlier calculations, we see that:

• Solid sphere has the highest acceleration: (5/7)g * sin(θ)
• Hollow sphere has a lower acceleration: (3/5)g * sin(θ)
• Disc has the lowest acceleration: (2/3)g * sin(θ)

Thus, the solid sphere will reach the bottom first, followed by the hollow sphere, and lastly the disc.

Final Answer

In conclusion, the solid sphere will take the least time to reach the bottom of the inclined plane, followed by the hollow sphere and then the disc. The order of time taken is:

• 1) Solid sphere
• 2) Hollow sphere
• 3) Disc