John is carrying a shovelful of snow. The center of mass of the 3.00 kg of snow he is holding is 15.0 cm from the end of the shovel. He is pushing down on the opposite end of the shovel with one hand and holding it up 30.0 cm from that end with his other hand. Ignore the mass of the shovel. Sarah says that his hand pushing down on the shovel must be exerting a greater force than the hand pushing up. James says it is just the reverse. Which one, if either, is correct?
If snow is downward, and upper hand is downward, then the lower hand is the sum of these. So lower hand is greater than upper hand.
To determine whether Sarah or James is correct, let's analyze the forces acting on the shovel.
1. Weight of the snow:
The snow has a mass of 3.00 kg, and its weight can be calculated using the formula:
Weight = mass * acceleration due to gravity
Weight = 3.00 kg * 9.8 m/s^2
Weight = 29.4 N
2. Force exerted by the hand pushing down:
The center of mass of the snow is 15.0 cm (or 0.15 m) from the end of the shovel. Since force is equal to weight multiplied by distance from the pivot point,
Force pushing down = Weight * Distance of center of mass from pivot point
Force pushing down = 29.4 N * 0.15 m
Force pushing down = 4.41 N
3. Force exerted by the hand pushing up:
The hand is holding the shovel up 30.0 cm (or 0.30 m) from the end. Therefore,
Force pushing up = Weight * Distance of center of mass from pivot point
Force pushing up = 29.4 N * 0.30 m
Force pushing up = 8.82 N
Comparing the forces, we can see that the hand pushing up exerts a greater force (8.82 N) compared to the hand pushing down (4.41 N). Thus, James is correct in saying that the hand pushing up must exert a greater force.
To determine which hand exerts a greater force on the shovel, we need to consider the torque acting on the shovel at the center of mass. Torque is the measure of the ability of a force to rotate an object about an axis and is given by the equation:
Torque = Force × Distance
In this scenario, the center of mass of the 3.00 kg of snow is 15.0 cm from the end of the shovel. The hand pushing down is also at a distance of 15.0 cm from the center of mass (but on the opposite side), and the hand pushing up is 30.0 cm away from the center of mass.
Now, let's calculate the torques exerted by each hand:
1. Torque exerted by the hand pushing down:
Torque_down = Force_down × Distance_down
2. Torque exerted by the hand pushing up:
Torque_up = Force_up × Distance_up
According to the problem, the weights of the snow on both sides of the shovel are equal, but we need to determine whether one hand exerts a greater force than the other.
Let's assume that the hand pushing down exerts a force F_down, and the hand pushing up exerts a force F_up.
Since the system is in equilibrium (not rotating), the torques exerted by each hand must be equal:
Torque_down = Torque_up
Now we can set up an equation:
Force_down × Distance_down = Force_up × Distance_up
We know that Distance_down = Distance_up = 15.0 cm = 0.15 m.
Therefore,
F_down × 0.15 m = F_up × 0.15 m
The distances cancel out, which means that:
F_down = F_up
This implies that both hands exert an equal amount of force on the shovel. Neither Sarah nor James is completely correct. Both hands are exerting the same force, but in opposite directions, resulting in no net torque and the shovel being in equilibrium.
So, the correct answer is that the forces exerted by both hands are equal.