## Problems

GSI = 0f a2'5 dt where a is the instantaneous acceleration of the head and t is the duration of the pulse. If the integrated value exceeds 1,000, severe injury is predicted to result. Consider a fall where the head of a person hits the ground with a vertical velocity of 8 m/s. The impact lasts 0.05 s and the impact force is uniform during the course of impact. Determine the value of GSI for this head impact. Answer: GSI = 16191.

Problem 7.4. If a person were not wearing a seat belt in a car when the car hit a wall or a large tree, the overall effect is that of a person hitting a massive wall with the velocity of the car before collision. In that sense a collision may be considered equivalent to falling from a height h onto a concrete sidewalk. Suppose the speed of the car before the collision is 90 km/h. Determine the height of a free fall that would give the same velocity before collision. Answer: H = 31.8 m.

Problem 7.5. In a study of hip fracture etiology, young healthy athletes weighing 70 kg performed voluntary sideways falls on a thick foam mattress. The mean value for the vertical impact velocity of the center of mass of a falling athlete was 2.75 m/s. Assuming that there was no rebound immediately after the impact, compute the vertical impulse due to the fall. Answer: £ = 192.5 N-s.

Problem 7.6. A uniform rod of mass m and length L can turn freely around point A (Fig. P.7.6). It is held in its highest position and is then allowed to fall. On reaching its lowest position, it encounters a fixed object. The object remains stationary while the rod rebounds. The coefficient of restitution e between the rod and the object is equal to 0.4. Find the impulse exerted on the rod. What is the rebound velocity at the site of the collision?

Figure P.7.6. The rotation of a uniform rod around - !

point A and the resulting impact with a stationary M '"'B object. _

Hint: To compute the velocity of the rod before collision, derive a differential equation for angular speed using conservation of angular momentum.

Problem 7.7. Consider sideways fall of an individual whose weight is 60 kg and height is 1.60 m. Represent the falling person as composed of two equal uniform rods AB (lower body) and BD (upper body), hinged together at B (hip joint) as shown in Fig. P.7.7. Both rods have mass m = 30 kg and length L = 0.8 m. Immediately before the impact, the point A (feet) was moving in the -ei direction with speed equal to 1.2 m/s. The angular velocities of the lower body and the trunk immediately before the impact were measured as -7.5 rad/s e3 and -4.8 rad/s e3. The angles the lower body and the trunk made with the e1 direction were 0 and ^/6, respectively. The lower body remained at rest on the ground immediately after the impact, whereas the upper body began rotating in the counterclockwise direction with angular velocity equal to 3.2 rad/s e3. Determine the impulse of the impact acting on the individual. Answer: £ = 24 e1 + 353 e2 (N-s).

Problem 7.8. A man hits a ball of radius R and mass m with a cylindrical rod of length L and mass M (Fig. P.7.8). Before the impact the ball had an initial velocity vo e2 and angular velocity Mo e3 as shown in the figure. The rod, on the other hand, was rotating around the stationary point A with angular velocity -Oo e3. The distance between the point of impact and point A is denoted as d. Assume the force of impulse to be perpendicular to the contact area. Determine the angular velocity O of the rod and velocity of the center of mass v of the ball immediately after the impact. use the following parameter values in your computations: M = 10 kg, L = 1 m, d = 0.8 m, Oo = 5 rad/s, m = 5 kg, R = 0.1 m, vo = 10 m/s, and the coefficient of restitution e = 0.8.

Figure P.7.7. A two-bar model of a sideways fall. The uniform rods AB and BD represent the upper and lower body, respectively.

Figure P.7.7. A two-bar model of a sideways fall. The uniform rods AB and BD represent the upper and lower body, respectively.

Figure P.7.8. The collision of a ball with a cylindrical rod that is free to rotate around point A.