Final answer:
To find the angle theta, we can use Newton's second law for rotational motion. The weight of the die applies a torque on the string which causes it to rotate. The torque can be calculated by multiplying the weight by the perpendicular distance from the axis of rotation to the string. Setting the torque equal to the moment of inertia multiplied by the angular acceleration, we can solve for theta.
Step-by-step explanation:
To find the angle theta, we can use Newton's second law for rotational motion. The weight of the die applies a torque on the string which causes it to rotate. The torque can be calculated by multiplying the weight by the perpendicular distance from the axis of rotation to the string. Since the string makes an angle theta with respect to the vertical, this perpendicular distance can be calculated as the length of the string multiplied by the sine of theta. Setting the torque equal to the moment of inertia multiplied by the angular acceleration, we can solve for theta.
The torque is given by: torque = weight * length * sin(theta)
The moment of inertia of the die can be calculated as the mass of the die multiplied by the square of the radius (assuming it is a solid sphere).
Setting the torque equal to the moment of inertia multiplied by the angular acceleration, we get: weight * length * sin(theta) = (2/5) * mass * radius^2 * angular acceleration
Simplifying the equation and solving for theta, we find: theta = arcsin((5 * weight * length) / (2 * mass * radius^2 * angular acceleration))