A person is sitting with one leg outstretched and stationary, so that it makes an angle of 30.0° with the horizontal, as the drawing indicates. The weight of the leg below the knee is 50.5 N, with the center of gravity located below the knee joint. The leg is being held in this position because of the force applied by the quadriceps muscle, which is attached 0.100 m below the knee joint (see the drawing). Obtain the magnitude of .
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To find the magnitude of the force applied by the quadriceps muscle, we can use the concept of torque.
The torque produced by a force is given by the equation:
Torque = Force * Lever Arm
In this case, the lever arm is the distance between the force and the axis of rotation. The force applied by the quadriceps muscle acts perpendicular to the leg, creating a torque that keeps the leg in position.
Since the leg is outstretched at an angle of 30.0° with the horizontal, the lever arm can be calculated using the formula:
Lever Arm = Distance * sin(angle)
Given that the distance between the force and the knee joint is 0.100 m, we can plug in the values into the formula:
Lever Arm = 0.100 m * sin(30.0°)
Lever Arm ≈ 0.050 m
Now, we can use the torque formula to find the magnitude of the force applied by the quadriceps muscle:
Torque = Force * Lever Arm
The weight of the leg below the knee is given as 50.5 N. Since the center of gravity is located below the knee joint, we can assume that the entire weight is acting at the knee joint.
Torque due to weight = Weight * Lever Arm
Torque due to weight = 50.5 N * 0.100 m
Torque due to weight = 5.05 Nm
Since the leg is not rotating, the torque produced by the quadriceps muscle must be equal and opposite to the torque due to weight.
So,
Torque due to muscle = -Torque due to weight
Force * Lever Arm = -Weight * Lever Arm
Force = (-Weight)
Therefore, the magnitude of the force applied by the quadriceps muscle is equal to the weight of the leg:
Magnitude of force = 50.5 N.
To obtain the magnitude of the force applied by the quadriceps muscle, we can use the concept of torque. Torque is the product of force and the perpendicular distance from the point of rotation.
In this case, the point of rotation is the knee joint, and the force applied by the quadriceps muscle creates a torque that acts in a clockwise direction.
Using the given information, we can calculate the torque and then determine the magnitude of the force.
1. Calculate the perpendicular distance:
The quadriceps muscle is attached 0.100 m below the knee joint. Since the leg is at an angle of 30.0° with the horizontal, we can use trigonometry to find the perpendicular distance.
Perpendicular Distance = 0.100 m * sin(30.0°)
2. Calculate the torque:
Torque = Force * Perpendicular Distance
3. Solve for the force:
Force = Torque / Perpendicular Distance
Substitute the values into the equation and calculate the force:
Force = Torque / (0.100 m * sin(30.0°))
Force = (Weight * Perpendicular Distance) / (0.100 m * sin(30.0°))
Given that the weight of the leg below the knee is 50.5 N, and we have already calculated the perpendicular distance, substitute these values and solve for force.
Force = (50.5 N * Perpendicular Distance) / (0.100 m * sin(30.0°))
Calculate the value of force using this equation and the given values, and you will obtain the magnitude of the force applied by the quadriceps muscle.