The length of a magnet is large compared to its width and breadth. The time period of its width and breadth. The time period of its oscillation in a vibration magnetometer is 2 s. The magnet is cut along its length into three equal parts and three parts are then placed on each other with their like poles together. The time period of this combination will be ?

To determine the time period of the combined magnet after cutting it into three equal parts and stacking them, we need to consider how the dimensions of the magnet affect its oscillation characteristics. The time period of oscillation for a magnet in a vibration magnetometer is influenced by its moment of inertia and the magnetic restoring torque acting on it.

Understanding the Basics of Magnetic Oscillation

The initial time period of oscillation for the original magnet is given as 2 seconds. The time period \( T \) of a magnet in a vibration magnetometer can be expressed using the formula:

T = 2π√(I/mgh)

Where:

• I is the moment of inertia of the magnet.
• m is the mass of the magnet.
• g is the acceleration due to gravity.
• h is the distance from the pivot point to the center of mass.

Effect of Cutting the Magnet

When the magnet is cut into three equal parts, each piece retains the same width and breadth but has one-third of the original length. The mass of each part will also be one-third of the original mass (assuming uniform density).

Next, when these pieces are stacked with their like poles together, the overall length is effectively restored to its original size, but the mass increases. The new mass \( m' \) of the combined magnet is now:

m' = 3 * (m/3) = m

Moment of Inertia Considerations

The moment of inertia for the original magnet can be expressed as:

I = (1/3) * m * l^2

Where \( l \) is the original length. For the three parts stacked together, the new moment of inertia \( I' \) when they are placed on top of each other is:

I' = 3 * (1/3) * m * (l/3)^2 = (1/3) * m * (l^2/3) = (1/9) * m * l^2

Calculating the New Time Period

Inserting the new moment of inertia into the time period formula:

T' = 2π√(I'/mgh)

Substituting for \( I' \):

T' = 2π√((1/9) * m * l^2 / (m * g * h))

We can simplify this expression:

T' = 2π√((1/9) * (l^2 / (g * h)))

The new time period can be derived from the original time period:

T' = T / 3 = 2 s / 3 = 0.67 s

Final Result

The time period of the stacked combination of the magnet will be approximately 0.67 seconds. This decrease in time period indicates that stacking the magnets increases their effective restoring force, allowing them to oscillate faster compared to the original single piece.