Determine the torques about the axis of rotation P produced by each of the four forces shown in the figure (Figure 1). All forces have magnitudes of 120 N and are exerted a distance of 2.0 m from P on some unshown object O.
Determine the torque about the axis of rotation P produced by F 1onO.
Determine the torque about the axis of rotation P produced by F 2onO.
Determine the torque about the axis of rotation P produced by F 3onO.
Determine the torque about the axis of rotation P produced by F 4onO.
To determine the torques about the axis of rotation P produced by each of the four forces, we can use the equation:
Torque (τ) = Force (F) x Lever Arm (r)
where the lever arm is the perpendicular distance from the axis of rotation to the line of action of the force.
Given that all forces have magnitudes of 120 N and are exerted at a distance of 2.0 m from P, we can calculate the torque for each force.
1. Torque produced by F1 on O:
The force F1 is perpendicular to the line OP, so the lever arm is the distance 2.0 m. The torque is calculated as:
τ1 = F1 x r1 = 120 N x 2.0 m = 240 N•m
2. Torque produced by F2 on O:
The force F2 is parallel to the line OP, so it does not contribute to the torque about P. The torque produced by F2 is zero.
3. Torque produced by F3 on O:
The force F3 is perpendicular to the line OP, so the lever arm is the distance 2.0 m. The torque is calculated as:
τ3 = F3 x r3 = 120 N x 2.0 m = 240 N•m
4. Torque produced by F4 on O:
The force F4 is parallel to the line OP, so it does not contribute to the torque about P. The torque produced by F4 is zero.
Therefore, the torques about the axis of rotation P produced by each force are:
τ1 = 240 N•m (clockwise)
τ2 = 0 N•m (no torque)
τ3 = 240 N•m (counterclockwise)
τ4 = 0 N•m (no torque)
To determine the torques about the axis of rotation P produced by each of the four forces, we need to use the formula for torque:
Torque = Force x Distance x sin(θ)
Where:
Force: the magnitude of the force
Distance: the perpendicular distance from the axis of rotation to the line of action of the force
θ: the angle between the force vector and the line connecting the axis of rotation and the point of application of the force
Now let's calculate the torque produced by each force.
1. To determine the torque about the axis of rotation P produced by F1 on O:
The force F1 is acting at a distance of 2.0 m from P. The angle θ is 90 degrees since F1 is perpendicular to the line connecting P and the point of application. Thus, sin(90 degrees) is equal to 1.
Therefore, the torque produced by F1 on O is:
Torque1 = Force x Distance x sin(θ)
= 120 N x 2.0 m x sin(90 degrees)
= 240 Nm
2. To determine the torque about the axis of rotation P produced by F2 on O:
Like F1, F2 is also acting at a distance of 2.0 m from P. However, F2 is at an angle of 120 degrees with respect to the line connecting P and the point of application. Thus, sin(120 degrees) is equal to √3/2.
Therefore, the torque produced by F2 on O is:
Torque2 = Force x Distance x sin(θ)
= 120 N x 2.0 m x sin(120 degrees)
= 240 Nm x (√3/2)
= 240√3 Nm
3. To determine the torque about the axis of rotation P produced by F3 on O:
Similar to F2, F3 is acting at a distance of 2.0 m from P. F3 is at an angle of 240 degrees with respect to the line connecting P and the point of application. However, since sin(240 degrees) is equal to -√3/2, the negative sign indicates the direction of the torque.
Therefore, the torque produced by F3 on O is:
Torque3 = Force x Distance x sin(θ)
= 120 N x 2.0 m x sin(240 degrees)
= 240 Nm x (-√3/2)
= -240√3 Nm
4. To determine the torque about the axis of rotation P produced by F4 on O:
Again, F4 is acting at a distance of 2.0 m from P. F4 is at an angle of 330 degrees with respect to the line connecting P and the point of application. Since sin(330 degrees) is equal to -1/2, the negative sign indicates the direction of the torque.
Therefore, the torque produced by F4 on O is:
Torque4 = Force x Distance x sin(θ)
= 120 N x 2.0 m x sin(330 degrees)
= 240 Nm x (-1/2)
= -120 Nm
To summarize:
- The torque produced by F1 on O is 240 Nm.
- The torque produced by F2 on O is 240√3 Nm.
- The torque produced by F3 on O is -240√3 Nm.
- The torque produced by F4 on O is -120 Nm.