Why is sin theta used in the torque formula while the work formula uses cosine theta?
The reason for using sin theta in the torque formula and cosine theta in the work formula is due to the way these formulas are derived from the principles of trigonometry and physics.
In the context of torque, which measures the rotational force being applied to an object, the formula is given as τ = rFsinθ, where τ represents torque, r is the distance from the axis of rotation to the point of application of the force, F is the magnitude of the force, and θ is the angle between the force vector and the line connecting the axis of rotation to the point where the force is applied.
The use of sin theta in the torque formula is due to the fact that torque is directly related to the component of the force vector perpendicular to the line connecting the axis of rotation to the point of application. This can be visualized using the concept of the lever arm, which is the distance from the axis of rotation to the line of action of the force. The component of the force vector that contributes to torque is the force perpendicular to the lever arm, which is given by Fsinθ.
On the other hand, in the context of work, which measures the energy transferred to or from an object by a force, the formula is given as W = Fdcosθ, where W represents work, F is the magnitude of the force, d is the distance over which the force is applied, and θ is the angle between the force vector and the direction of displacement.
The use of cosine theta in the work formula is due to the fact that work is directly related to the component of the force vector that is parallel to the direction of displacement. This component of the force is given by Fcosθ.
Therefore, sin theta is used in the torque formula because torque is related to the component of the force perpendicular to the lever arm, while cosine theta is used in the work formula because work is related to the component of the force parallel to the direction of displacement. The choice of sin or cosine depends on the specific physical quantity being calculated and the geometry of the situation.