Halliday, Resnick & Walker — Kinetic Energy and Work: Problem 74

Hip fracture and body mass index. Hip fracture due to a fall is a chronic problem, especially with older people and people subject to seizures. One research focus is on the correlation between fracture risk and weight, specifically, the body mass index (BMI). That index is defined as \( m/h^2 \), where \( m \) is the mass (in kilograms) and \( h \) is the height (in meters) of a person. Is a person with a higher BMI more or less likely to fracture a hip in a fall on a floor? One way to measure the fracture risk is to measure the amount of energy absorbed as the hip impacts the floor and any covering in a sideways fall. During the impact and compression of the floor and covering, the force from the hip does work on the floor and covering. A larger amount of work implies a smaller amount of energy left to fracture the hip. In an experiment, a participant is held horizontally in a sling with the left hip 5.0 cm above a force plate with a floor covering. When the participant is dropped, measurements are made of the force magnitude \( F \) on the plate during impact and the plate’s deflection \( d \). Figure 7.35 gives idealized plots for two participants. For A: \( m = 55.0 \text{ kg} \), \( h = 1.70 \text{ m} \), peak force \( F_A = 1400 \text{ N} \), and maximum plate deflection \( d_A = 2.00 \text{ cm} \). For B: \( m = 110 \text{ kg} \), \( h = 1.70 \text{ m} \), \( F_{B2} = 1600 \text{ N} \), \( d_{B2} = 6.00 \text{ cm} \), and intermediate force \( F_{B1} = 500 \text{ N} \) at deflection \( d_{B1} = 4.00 \text{ cm} \). What is the BMI for (a) (lighter) participant A and (b) (heavier) participant B? (c) Which participant experiences the greater peak force from the plate, the lighter one or the heavier one? How much energy is absorbed by the plate and covering (how much work is done on the plate) for (d) participant A and (e) participant B? What is the absorbed energy per unit mass of (f) participant A and (g) participant B? (h) Do the results indicate higher plate absorption (and thus lower fracture risk for the participant) for the higher BMI or lower BMI? (i) Does this correlate with the peak force results?