A young girl sits at the edge of a dock by the bay, dipping her feet in the water. At the instant shown in the figure , she holds her lower leg stationary with her quadriceps muscle at an angle of 39 degree with respect to the horizontal.
Use the information given in the figure, plus the fact that her lower leg has a mass of 3.3 kg, to determine the magnitude of the force, F, exerted on the lower leg by the quadriceps.
F = (3.3 kg)(9.8 m/s^2)sin(39°) = 28.9 N
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To determine the magnitude of the force, F, exerted on the lower leg by the quadriceps, we need to examine the forces acting on the leg.
First, it's important to note that the force exerted by the quadriceps is in equilibrium with the gravitational force acting on the lower leg. This is because the leg is held stationary and not moving vertically or horizontally.
Now, let's break down the forces acting on the lower leg:
1. Gravitational force (mg): The lower leg has a mass of 3.3 kg, so its weight can be calculated using the equation: weight = mass * acceleration due to gravity. Assuming the acceleration due to gravity is 9.8 m/s^2, the gravitational force can be calculated as: weight = 3.3 kg * 9.8 m/s^2.
2. Force exerted by the quadriceps (F): This is the force we want to determine, and it is directed along the quadriceps muscle at an angle of 39 degrees with respect to the horizontal.
Since the leg is held stationary, the vertical component of the force exerted by the quadriceps must be equal in magnitude and opposite in direction to the gravitational force:
F * sin(39 degrees) = weight
Solving this equation will give us the desired magnitude of the force, F.
To determine the magnitude of the force, F, exerted on the lower leg by the quadriceps, we can analyze the forces acting on the leg and use Newton's second law of motion (F = ma) to solve for the force.
First, let's identify the forces acting on the lower leg:
1. The weight of the lower leg.
2. The normal force exerted on the lower leg by the dock.
3. The force exerted on the lower leg by the quadriceps (F).
Since the lower leg is stationary, the forces in the horizontal direction must be balanced. The force exerted by the quadriceps (F) must counteract the horizontal component of the weight of the lower leg.
To determine the horizontal component of the weight, we need to find the angle between the leg and the vertical. Since the angle between the quadriceps and the horizontal is given as 39 degrees, the angle between the leg and the vertical will be the complement of 39 degrees, which is 90 - 39 = 51 degrees.
Now, let's calculate the horizontal component of the weight:
Horizontal component of weight = weight of the leg * cos(angle between leg and vertical)
The weight of the leg can be calculated using the formula:
Weight = mass * acceleration due to gravity
Weight of the leg = 3.3 kg * 9.8 m/s^2 (acceleration due to gravity)
Substituting the values, we find the weight of the leg.
Next, we can calculate the horizontal component of the weight:
Horizontal component of weight = weight of the leg * cos(51 degrees)
Now, since the lower leg is stationary, the force exerted by the quadriceps (F) must equal the horizontal component of the weight. So, we can write the equation:
F = horizontal component of weight
Substituting the calculated value of the horizontal component of the weight, we can find the magnitude of the force, F, exerted on the lower leg by the quadriceps.