A sturdy wooden board, 4.0 meters long and with a mass of 32 kg, rests on two supports (labeled L and R) placed 1.2 m from each end as shown in the figure below.
(a) Suppose that a man with a mass of 60 kg stands at the center of the board (point a). What are the upward forces on the board from each support?
(b) Now the man stands 1.0 m from one end of the board (point b). What are the upward forces from each support now?
(c) The man edges gradually toward the end of the board. At what distance from the end does the board finally tip?
1. draw the figure, with forces.
2. Sum the vertical forces, and set to zero.
3. Sum moments about either end.
You will have the solution.
To find the upward forces on the board from each support, we need to consider the torques acting on the board.
(a) When the man stands at the center of the board (point a), the board is in equilibrium, which means the net torque acting on the board must be zero. The torque is given by the equation τ = F * d, where F is the force and d is the distance from the pivot point.
Since the board is balanced and the man is at the center, the net torque equation becomes:
Torque from left support = Torque from right support
The torque from each support is calculated as:
Torque from support = Force from support * distance from the support
Let's calculate the forces from each support. Due to symmetry, the forces on both sides will be equal.
Distance from the left support = 1.2 m
Distance from the right support = 4.0 m - 1.2 m = 2.8 m
Let F_L be the force from the left support and F_R be the force from the right support.
The equation for the torques becomes:
F_L * 1.2 m = F_R * 2.8 m
Now let's consider the vertical forces.
The sum of the upward forces must equal the downward force acting on the board, which is the weight of the board and the man combined (32 kg + 60 kg) * 9.8 m/s^2.
F_L + F_R = (32 kg + 60 kg) * 9.8 m/s^2
Now, we have two equations:
F_L * 1.2 m = F_R * 2.8 m
F_L + F_R = (32 kg + 60 kg) * 9.8 m/s^2
By solving these equations, we can find the upward forces on the board from each support.
(b) When the man stands 1.0 m from one end of the board (point b), the process is similar. The main difference is that the distances from the supports change.
Distance from the left support = 1.2 m + 1.0 m = 2.2 m
Distance from the right support = 4.0 m - 2.2 m = 1.8 m
Repeating the same calculation process, you can find the upward forces on the board from each support.
(c) To find the distance from the end at which the board finally tips, we need to determine the conditions for tipping. Tipping occurs when the net torque becomes sufficiently large to overcome the stability of the board.
As the man edges towards the end of the board, the distance from the left support decreases, and the distance from the right support increases.
Eventually, there will be a point where the torque from the right support is large enough to tip the board. At this point, the net torque equation becomes:
F_L * d_L = F_R * d_R
where d_L is the distance from the left support and d_R is the distance from the right support.
Note that the sum of the upward forces is still governed by:
F_L + F_R = (32 kg + 60 kg) * 9.8 m/s^2
You can use these equations to determine the distance from the end at which the board finally tips by solving for d_L and d_R.