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An 80 kg solid sphere with a 0.1 m radius is suspended by a vertical wire attached to the ceiling of a room. If the sphere is given a twist and released, it to oscillates with a period of 5 s. How much torque is required to twist and hold the sphere at an angle of 1.5 radians

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Answer:

The torque required is 0.024Nm

Explanation:

Solution:

Data Given;

Mass of sphere = 80kg

Radius of sphere = 0.1m

Time = 5s

Angle = 1.5radians

To calculate the torque is required to twist and hold the sphere, we use the formula;

Torque = Inertia * angular acceleration -----------------------------1

where,

Inertia = 1/2 mr²---------------------------------------------------------------2

Substituting into the formula, we have

           = 1/2 * 80*0.1²

           = 1/2 * 0.8

           = 0.4kgm²

Angular acceleration = angular velocity/time----------------------3

Where angular velocity = ∅/time

Substitution into equation 3, we have

                                   = 1.5/5²

                                   = 1.5/25

                                   = 0.06rad/s²

Coming back to equation 1,

Substituting the values obtained into equation 1, we have

Torque = Inertia * angular acceleration

            = 0.4 * 0.06

            = 0.024Nm

Answer:

Torque = 0.7575 N.m

Explanation:

We are given;

Mass; m = 80kg

Radius = 0.1m

Period; T = 5s

Angle; θ = 1.5 radians

In simple harmonic motion, period T is given as;

T = 2π√(I/k)

Where,

I is rotational inertia of the object

k is torsion constant of the wire

Now, The rotational inertia has a formula;

I = (2/5)mr²

Where m is mass and r is radius.

Thus,

I = (2/5) x 80 x 0.1² = 0.32 kg.m²

Thus, we can now solve for k in the period equation ;

T = 2π√(I/k)

Make k the subject ;

k = 4π²I/(T²)

k = (4π² x 0.32)/(5²)

k = 0.505 N.m/rad

Now, the torque required to twist and hold is given by;

τ = kθ

Now, we know that k = 0.505 and θ = 1.5 rads

Thus, τ = 0.505 x 1.5 = 0.7575 N.m

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