Find the net torque (magnitude and direction) produced by the forces F1 (F1 = 17.3 N) and F2 (F2 = 26.1 N) about the rotational axis shown in the drawing. The forces are acting on a thin rigid rod, and the axis is perpendicular to the page. (x = 0.323 m, y = 1.021 m, = 28.6°)
To find the net torque produced by the forces F1 and F2 about the given rotational axis, we need to use the equation for torque:
τ = r × F
where τ is the torque, r is the position vector from the axis of rotation to the point of application of the force, and F is the applied force.
First, let's write down the values given in the problem:
F1 = 17.3 N (magnitude of the force F1)
F2 = 26.1 N (magnitude of the force F2)
x = 0.323 m (x-component of the position vector)
y = 1.021 m (y-component of the position vector)
θ = 28.6° (angle between the position vector and the axis of rotation)
Now, we need to find the position vectors r1 and r2 for forces F1 and F2, respectively. The position vector is the vector that connects the axis of rotation to the point of application of the force. In this case, since the axis of rotation is perpendicular to the page, the position vectors will have only x and y components.
For force F1:
r1 = (x1, y1) = (0.323 m, 1.021 m)
For force F2:
r2 = (x2, y2) = (0.323 m, 1.021 m)
Next, we calculate the cross product of each position vector with its corresponding force to find the torque:
τ1 = r1 × F1
τ2 = r2 × F2
The cross product of two vectors in 2D is calculated as the magnitude of the vectors multiplied by the sine of the angle between them.
For torque τ1:
|τ1| = |r1| * |F1| * sin(θ)
= √(x1^2 + y1^2) * F1 * sin(θ)
For torque τ2:
|τ2| = |r2| * |F2| * sin(θ)
= √(x2^2 + y2^2) * F2 * sin(θ)
Now we can substitute the given values and calculate the magnitude of the torque for each force:
|τ1| = √(0.323^2 + 1.021^2) * 17.3 N * sin(28.6°)
|τ2| = √(0.323^2 + 1.021^2) * 26.1 N * sin(28.6°)
Finally, to find the net torque, we add the torques produced by each force, considering their directions:
Net torque = |τ1| + |τ2|
Since the problem does not specify the direction, you would need to include the direction separately. Torque is a vector quantity that depends on the right-hand rule and follows the convention of positive direction depending on the axis of rotation.
Hence, the net torque has a magnitude given by |τ1| + |τ2| and its direction can be determined using the right-hand rule or by considering the signs and directions during the calculations.