Question

Find the forces exerted by the two supports of a 4.0-m, 50-kg uniform cantilever (diving board) when a 8.5-kg duck stands at the opposite end, as shown in the figure below.

Answer 1

We need to sum the forces about a single point.

The sketch is using the location of F2 as a reference for dimensions, so we'll use that point.

Let's make our notation a little easier,

This notation is nice.

Point A is where F1 is applied. B is where F2 is applied. C is the center of gravity. D is where the duck is.

Let's first define our coordinate system. Forces going upwards are positive y-direction, e.g. Force 2 has a positive value.

Now, we can sum our forces about B in the y-direction. We need not worry about the x-direction, since no forces are acting in this direction.

Since the diving board is not moving, Newton's second law tells us that the sum of the forces must equal zero. \[\sum F_y = 0\]

Realizing the forces, \[\sum F_y = 0 = (-F_1) + F_2 - w_1 - w_2\]

We know \(w_1\) to be 40*g and \(w_2\) to be 8.5*g, but we don't know F_1 nor F_2.

We need another equation to satisfy the degrees of freedom of the equations.

Let's sum the torques (or moments) about point B. Let's define the direction of positive torque. Clockwise rotations about point B will be considered positive.

From Newton's Second Law, we know that the sum of the torques must equal zero, since the board is not moving. \[\sum \tau = 0\]

Realizing the torques, \[\sum \tau = 0 = -(F_1 \cdot 0.8) + w_1 \cdot 1.2 + w_2 \cdot 3.2\]

We now have two equations and two unknowns.

We can solve for F_1 and F_2.