A person exerts a force of 45.0 N on the end of a door 84.0 cm wide. What is the magnitude of the torque if the force is exerted:
a) perpendicular to the door, and
b) at a 60 degree angle to the face of the door?
45 * .84
45 * .84 * sin 60
Thanks!
To calculate the torque exerted on the door, we need to multiply the force applied by the lever arm, which is the perpendicular distance from the point of rotation to the line of action of the force.
a) If the force is exerted perpendicularly to the door:
The lever arm is equal to the width of the door.
Given:
Force (F) = 45.0 N
Width of the door (d) = 84.0 cm
First, we need to convert the width of the door from centimeters to meters:
d = 84.0 cm * 0.01 m/cm = 0.84 m
Now, we can simply calculate the torque (τ) using the equation:
τ = F * d
τ = 45.0 N * 0.84 m
τ = 37.8 N·m
Therefore, the magnitude of the torque exerted when the force is perpendicular to the door is 37.8 N·m.
b) If the force is exerted at a 60-degree angle to the face of the door:
In this case, we need to find the component of the force that is perpendicular to the door and then calculate the torque using that component.
The perpendicular component of the force (F_perpendicular) is given by:
F_perpendicular = F * sin(θ), where θ is the angle between the force and the door.
Given:
Force (F) = 45.0 N
Angle (θ) = 60 degrees
Calculating the perpendicular component of the force:
F_perpendicular = 45.0 N * sin(60°)
F_perpendicular = 45.0 N * 0.866
F_perpendicular = 38.97 N
Now, we can calculate the torque (τ) using the perpendicular component of the force and the width of the door:
τ = F_perpendicular * d
τ = 38.97 N * 0.84 m
τ = 32.71 N·m
Therefore, the magnitude of the torque exerted when the force is at a 60-degree angle to the face of the door is 32.71 N·m.
To find the magnitude of the torque in this situation, we need to use the formula: Torque = Force * lever arm. The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force.
a) If the force is exerted perpendicular to the door:
In this case, the line of action of the force coincides with the lever arm. Since the force exerted is 45.0 N and the door is 84.0 cm wide, which is equal to 0.84 m, the torque can be calculated as:
Torque = Force * lever arm = 45.0 N * 0.84 m = 37.8 N·m.
b) If the force is exerted at a 60 degree angle to the face of the door:
In this case, the line of action of the force is not perpendicular to the door, so we need to determine the perpendicular component of the force. To find the perpendicular component, we multiply the force by the sine of the angle between the force and the lever arm.
Perpendicular component of force = Force * sin(angle) = 45.0 N * sin(60°) = 45.0 N * 0.866 = 38.97 N.
Now, we need to determine the lever arm. The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force. Since the force is at a 60 degree angle to the face of the door, we can either use the height of the door or calculate the perpendicular distance using trigonometry.
If we use the height of the door, let's assume it is h meters. Then, the lever arm is h/2, since the force is acting at half the height of the door. However, since the height of the door is not given, we cannot calculate the exact lever arm.
Alternatively, we can calculate the perpendicular distance using trigonometry. The lever arm can be determined by multiplying the width of the door by the cosine of the angle between the force and the lever arm.
Lever arm = Door width * cos(angle) = 0.84 m * cos(60°) = 0.84 m * 0.5 = 0.42 m.
Now that we have the perpendicular component of the force and the lever arm, we can calculate the torque:
Torque = Perpendicular component of force * lever arm = 38.97 N * 0.42 m = 16.37 N·m.
Therefore, the magnitude of the torque when the force is exerted at a 60 degree angle to the face of the door is approximately 16.37 N·m.