You are taking care of two small children, Sarah and Rachel, who are twins. On a nice cold, clear day you decide to take them ice skating on Lake of the Isles. To travel across the frozen lake, you have Sarah hold your hand and Rachel's hand. The three of you form a straight line as you skate, and the two children just glide. Sarah must reach up at an angle of 60 degrees to grasp your hand, but she grabs Rachel's hand horizontally. Since the children are twins, they are the same height and the same weight, 50 lbs. To get started you accelerate at 2.0 m/s2. You are concerned about the force on the children's arms which might cause shoulder damage. So you calculate the force Sarah exerts on Rachel's arm, and the force you exert on Sarah's other arm. You assume that the frictional forces of the ice surface on the skates are negligible.

Force exerted by Sarah on Rachel's arm = 45.36 N

Force exerted by you on Sarah's arm = 118.42 N

Explanation:

To calculate the force exerted on Sarah's hand that your are holding, we need to use the following equation:

Force = mass * acceleration

Since you have to accelerate both the twins, each weighing in at 50 lbs or 22.6796 kg, at a rate of 2.0 m/s^2, we get:

Force = (22.6796 * 2) * 2

Force = 90.718 N   (in the horizontal direction)

Since this is the horizontal force needed, and your arm is 40° to Sarah's, we have the following relation:

Horizontal Force = Total force on arm * Cos(40°)

90.718 = Total force on arm * Cos(40°)

Total force on Sarah's arm = 118.424 N     (this is the arm you're holding)

The force exerted by Sarah on Rachel is half of the horizontal force (90.718 N) needed to get them both accelerating. This is because their weight is the same. Thus,

Force Sarah exerts on Rachel's arm = 90.718 / 2 = 45.36 N

see earlier