To analyze the behavior of the disc under the given sinusoidal torque, we need to break down the problem step by step. The torque applied is described by the equation Torque = 100sin(π/10 * t), where t is in seconds. The moment of inertia of the disc is provided as 10 kgm². Let's evaluate each statement one by one.
Understanding Torque and Angular Motion
Torque is the rotational equivalent of linear force. It causes an object to rotate around an axis. The relationship between torque (τ), moment of inertia (I), and angular acceleration (α) is given by the equation:
τ = I * α
From this, we can derive the angular acceleration:
α = τ / I
Calculating Angular Acceleration
Substituting the expression for torque into the equation gives us:
α = (100sin(π/10 * t)) / 10 = 10sin(π/10 * t)
This means the angular acceleration varies sinusoidally with time, oscillating between positive and negative values as the sine function oscillates.
Evaluating Each Statement
- Statement A: The disc rotates in the anticlockwise direction between 0 and 10s and in the clockwise direction between 10 and 20s, i.e., it oscillates between the two.
This statement is true. Between 0 and 10 seconds, the torque is positive (since sin(π/10 * t) is positive), causing the disc to rotate anticlockwise. After 10 seconds, the torque becomes negative, leading to clockwise rotation.
- Statement B: The angular velocity of the disc increases between 0 and 10s and decreases between 10 and 20s; but always remains in the anticlockwise direction.
This statement is false. While the angular velocity does increase between 0 and 10 seconds, it does not remain in the anticlockwise direction after 10 seconds, as it starts to rotate clockwise due to negative torque.
- Statement C: The maximum angular velocity of the disc is 200/π rad/s at t=10s.
This statement is incorrect. To find the maximum angular velocity, we need to integrate the angular acceleration over time. The maximum angular velocity occurs when the torque is at its peak, which is at t = 5 seconds (where sin(π/10 * t) = 1). The angular velocity at this point can be calculated by integrating the angular acceleration from 0 to 5 seconds.
- Statement D: The maximum angular velocity of the disc is 100 rad/s at t=5s.
This statement is also false. The maximum angular velocity can be calculated by integrating the angular acceleration over the interval from 0 to 5 seconds. The integral of 10sin(π/10 * t) from 0 to 5 seconds gives us the angular velocity at that point, which is not equal to 100 rad/s.
Calculating Maximum Angular Velocity
To find the maximum angular velocity, we integrate the angular acceleration:
ω(t) = ∫α dt = ∫10sin(π/10 * t) dt
Performing this integral from 0 to t gives:
ω(t) = -100/π * cos(π/10 * t) + C
At t = 0, ω(0) = 0, so C = 100/π. Thus:
ω(t) = -100/π * cos(π/10 * t) + 100/π
The maximum occurs when cos(π/10 * t) = -1, which gives us:
ω_max = 200/π rad/s at t = 10 seconds.
Final Thoughts
In summary, the true statements are:
- A) True
- B) False
- C) True
(with the correct understanding of the timing)
- D) False
Understanding the dynamics of rotational motion and how torque influences angular velocity is crucial in physics. Keep practicing these concepts, and you'll find them becoming clearer over time!
