A person is doing leg lifts with 3.0-kg ankle weights. She is sitting in a chair with her legs bent at a right angle initially. The quadriceps muscles are attached to the patella via a tendon; the patella is connected to the tibia by the patellar tendon, which attaches to bone 10.0 cm below the knee joint. Assume that the tendon pulls at an angle of 20.0 with respect to the lower leg, regardless of the position of the lower leg. The lower leg has a mass of 5.0 kg and its centre of gravity is 22 cm below the knee. The ankle weight is 41 cm from the knee. If the person lifts one leg, find the force exerted by the patellar tendon to hold the leg at an angle of 90.0 with respect to the vertical (i.e. to hold the leg horizontal).

Question

A person is doing leg lifts with 3.0-kg ankle weights. She is sitting in a chair with her legs bent at a right angle initially. The quadriceps muscles are attached to the patella via a tendon; the patella is connected to the tibia by the patellar tendon, which attaches to bone 10.0 cm below the knee joint. Assume that the tendon pulls at an angle of 20.0 with respect to the lower leg, regardless of the position of the lower leg. The lower leg has a mass of 5.0 kg and its centre of gravity is 22 cm below the knee. The ankle weight is 41 cm from the knee. If the person lifts one leg, find the force exerted by the patellar tendon to hold the leg at an angle of 90.0 with respect to the vertical (i.e. to hold the leg horizontal).

I don't know how to start this question, would I use torque=rFsinTHETA? then rearrange to get F? I'm not sure how to find the torque though.

How do I start this question?

Answer 1

To solve this question, you are correct in thinking that you can use the torque equation:

Torque = r * F * sin(θ)

where:
- Torque is the rotational force (in Nm)
- r is the distance from the pivot point to where the force is applied (in meters)
- F is the force applied (in newtons)
- θ is the angle between the force and the lever arm (in degrees)

In this case, the pivot point is the knee joint, and the force is being exerted by the patellar tendon.

To find the torque exerted by the ankle weight on the leg, you need to calculate the force applied by the ankle weight and the distance from the pivot point (knee joint) to where the force is applied.

First, calculate the force exerted by the ankle weight using Newton's second law:

Force = mass * acceleration due to gravity

The mass in this case is the combined mass of the ankle weight and the lower leg (3.0 kg + 5.0 kg), and the acceleration due to gravity is 9.8 m/s².

Next, calculate the distance from the pivot point (knee joint) to where the force is applied (ankle weight):

Distance = 0.41 m + 0.10 m

Now you have the force and distance required to calculate the torque. However, the angle between the force and the lever arm is not given directly. Instead, you are told that the tendon pulls at an angle of 20.0 degrees with respect to the lower leg at all times, regardless of its position.

To find the angle between the force and the lever arm (θ), you need to subtract the angle mentioned from 90 degrees (since the leg is held at an angle of 90 degrees to the vertical).

θ = 90 degrees - 20.0 degrees

Now you have all the values needed to calculate the torque:

Torque = Distance * Force * sin(θ)

Finally, the force exerted by the patellar tendon can be found by rearranging the torque equation:

Force = Torque / (Distance * sin(θ))

By substituting the values obtained above into this equation, you can find the force exerted by the patellar tendon to hold the leg at an angle of 90 degrees with respect to the vertical.