A cat walks along a uniform plank that is 4.00 m long and has a mass of 7.00 kg. The plank is supported by two sawhorses, one 0.44 m from the left end of the board and the other 1.60 m from its right end. When the cat nears the right end, the plank just begins to tip. If the cat has a mass of 5.4 kg, how close to the right end of the two-by-four can it walk before the board begins to tip?
I am usually good at torque equilibrium, but I am not sure how to solve this one. How do you know when it is just about to tip?
To find out when the plank is just about to tip, we need to determine the torque exerted by the cat and compare it to the torque exerted by the plank itself.
Let's start by calculating the torque exerted by the cat. Torque (τ) is equal to the force (F) applied perpendicular to the lever arm (r). In this case, the force is the weight of the cat, and the lever arm is the distance between the cat's position and the pivot point (sawhorse).
The weight of the cat can be calculated using the formula: weight = mass * acceleration due to gravity. So, the weight of the cat is (5.4 kg * 9.8 m/s^2) = 52.92 N.
Since the cat is located on the right end of the plank, the distance between the cat's position and the right sawhorse (lever arm) is 1.60 m.
Now, the torque (τ) exerted by the cat can be calculated as τ = F * r = (52.92 N) * (1.60 m) = 84.67 N·m.
Next, let's calculate the torque exerted by the plank. Since the plank is uniform, we can consider its weight as acting at its center of gravity, which is at the midpoint of the plank (2.00 m from either side).
The weight of the plank can be calculated using the formula: weight = mass * acceleration due to gravity. So, the weight of the plank is (7.00 kg * 9.8 m/s^2) = 68.6 N.
Now, we need to find the effective lever arm for the plank's weight. Since the plank is supported by sawhorses, the weight exerts a torque about both sawhorses. The torque about the left sawhorse is (68.6 N) * (0.44 m) = 30.184 N·m. The torque about the right sawhorse is (68.6 N) * (1.60 m) = 109.76 N·m.
To determine when the plank is just about to tip, the torques exerted by the cat and the plank need to be equal. Therefore, we can set up the equation:
Torque by cat = Torque by plank
84.67 N·m = 30.184 N·m (around the left sawhorse) + 109.76 N·m (around the right sawhorse)
Now, we can solve for the distance from the right end of the plank that the cat can walk before the board begins to tip.
Let's call this distance "x." So, the distance between the cat's position and the right sawhorse is (1.60 m - x).
Now, we rewrite the torque equation in terms of "x":
(52.92 N) * (1.60 m - x) = 30.184 N·m + 109.76 N·m
Simplifying the equation, we get:
84.672 N·m - 52.92 N·m * x = 30.184 N·m + 109.76 N·m
Combining like terms:
-52.92 N·m * x = 55.272 N·m
Now, we can solve for "x" by dividing both sides of the equation by -52.92 N·m:
x = (55.272 N·m) / (52.92 N·m)
x ≈ 1.04 m
Therefore, the cat can walk up to approximately 1.04 meters from the right end of the plank before it begins to tip.