trap door of length and width 1.65 m is held open at an angle of 65.0 degree with respect to the floor. A rope is attached to the raised edge of the door and fastened to the wall behind the door in such a position that the rope pulls perpendicularly to the trap door. If the mass of the trap door is 16.8 kg, what is the torque exerted on the trap door by the rope?
To calculate the torque exerted on the trap door by the rope, we first need to determine the force exerted by the rope on the door.
1. Calculate the component of the force perpendicular to the door:
- The force exerted by the rope can be divided into two components: one parallel to the door and one perpendicular.
- Since the rope pulls perpendicularly to the trap door, the entire force is perpendicular to the door.
- Therefore, the force exerted by the rope on the door is equal to the tension in the rope.
2. Calculate the tension in the rope:
- The tension in the rope can be calculated using the weight (force due to gravity) of the door.
- The weight of the door can be calculated using the mass of the door and the acceleration due to gravity (`g`).
- Assuming `g` is approximately 9.8 m/s^2, the weight of the door (`W`) can be calculated as follows:
W = mass * g
3. Calculate the torque exerted by the rope on the door:
- Torque is defined as the product of force (`F`) and the perpendicular distance (`r`) from the axis of rotation (the hinge) to the line of action of the force.
- Since the force is perpendicular to the trap door, the perpendicular distance is the distance from the hinge to the point where the force is applied.
- In this case, the force is applied at the raised edge of the door, and we need to find the perpendicular distance from the hinge to that point.
- The perpendicular distance can be calculated using basic trigonometry: `r = length * sin(angle)`
- Finally, the torque (`τ`) exerted by the rope on the door is given by the formula: τ = F * r
Now, let's calculate the torque exerted on the trap door by the rope.
Given:
- Length of the trap door (length) = 1.65 m
- Angle with respect to the floor (angle) = 65.0 degrees
- Mass of the trap door (mass) = 16.8 kg
- Acceleration due to gravity (g) = 9.8 m/s^2
Step 1: Calculate the weight of the door.
W = mass * g
W = 16.8 kg * 9.8 m/s^2
Step 2: Calculate the perpendicular distance (r) from the hinge to the point where the force is applied.
r = length * sin(angle)
Step 3: Calculate the torque exerted by the rope on the door.
τ = F * r
Now you can substitute the values calculated in Step 1 and Step 2 into Step 3 to calculate the torque.
To find the torque exerted on the trap door by the rope, we need to first understand the concept of torque and the formula to calculate it.
Torque is a measure of how effectively a force can rotate an object around a pivot point. It depends on the magnitude of the force, the distance between the point of application of the force and the pivot point (radius), and the angle between the force and the direction perpendicular to the radius.
The formula to calculate torque is:
Torque = Force x Radius x sin(θ)
Where:
- Torque is the rotational force (in N.m or Newton-meter)
- Force is the applied force on the object (in Newtons)
- Radius is the perpendicular distance between the pivot point and the point of application of the force (in meters)
- θ (theta) is the angle between the applied force and the direction perpendicular to the radius (in radians)
Given data:
- Length (L) and width (W) of the trap door = 1.65 m
- Angle (θ) between the door and the floor = 65.0 degrees
- Mass (m) of the trap door = 16.8 kg
To calculate the torque, we need to find the force and the radius.
1. Force: The force can be determined since the rope pulls perpendicularly to the trap door, which means it is equal to the weight of the trap door.
Force = Mass x Gravity
Now, let's calculate the force:
Force = 16.8 kg x 9.8 m/s² (acceleration due to gravity)
Force = 164.64 N
2. Radius: The radius is the perpendicular distance between the point of rotation (pivot point) and the point where the force is applied. In this case, the pivot point is the hinged edge of the door.
To calculate the radius, we need to find the distance from the point of rotation (hinged edge of the door) to the force applied by the rope. Since the rope is attached to the raised edge of the trap door, the distance from the hinged edge to the raised edge is half the length of the door.
Radius = Length/2
Now, substitute the given value to find the radius:
Radius = 1.65 m / 2
Radius = 0.825 m
3. Angle (θ): The angle between the applied force and the direction perpendicular to the radius is 90 degrees since the force is pulling perpendicularly.
Now, let's calculate the torque using the formula:
Torque = Force x Radius x sin(θ)
Substituting the known values:
Torque = 164.64 N x 0.825 m x sin(90 degrees)
Torque = 135.6468 N.m (rounded to the appropriate significant figures)
Therefore, the torque exerted on the trap door by the rope is approximately 135.65 N.m.