The wheels, axle, and handles of a wheelbarrow weigh W = 53 N. The load chamber and its contents weigh WL = 547 N. The drawing shows these two forces in two different wheelbarrow designs. To support the wheelbarrow in equilibrium, the man’s hands apply a force to the handles that is directed vertically upward. Consider a rotational axis at the point where the tire contacts the ground, directed perpendicular to the plane of the paper. Find the magnitude of the man’s force for both designs.
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To find the magnitude of the man's force in both designs, we can use the principle of torque equilibrium.
In rotational equilibrium, the sum of torques acting on an object must be zero. Torque is calculated by multiplying the force applied at a distance from the axis of rotation.
Let's consider Design 1 first:
The torque due to the weight of the wheels, axle, and handles can be calculated by taking the weight (W = 53 N) and multiplying it by the distance between the point where the tire contacts the ground and the point where the man's hands apply the force. Let's call this distance d1.
Similarly, the torque due to the weight of the load chamber and its contents can be calculated by taking the weight (WL = 547 N) and multiplying it by the distance between the point where the tire contacts the ground and the center of mass of the load chamber and its contents. Let's call this distance d2.
Since the wheelbarrow is in equilibrium, the sum of these torques must be zero:
Torque due to the weight of wheels, axle, and handles + Torque due to the weight of load chamber and contents = 0.
Mathematically, this can be written as:
(W)(d1) + (WL)(d2) = 0.
Now, let's consider Design 2:
The weight of the wheels, axle, and handles, as well as the weight of the load chamber and its contents, act at the same distance from the axis of rotation (the point where the tire contacts the ground). Therefore, the torques due to these forces will be zero.
In this case, the man's force will be the only force creating a torque. Let's call the distance between the point where the tire contacts the ground and the point where the man's hands apply the force d3.
To find the magnitude of the man's force, we can use the same principle of torque equilibrium:
(W)(d3) = 0.
Since the torque due to the man's force is the only torque in Design 2, this equation holds.
To summarize:
- In Design 1, the magnitude of the man's force can be found by solving the equation (W)(d1) + (WL)(d2) = 0.
- In Design 2, the magnitude of the man's force can be found by solving the equation (W)(d3) = 0.
Please provide the values of distances (d1, d2, and d3) for each design to calculate the magnitude of the man's force in both designs.
To find the magnitude of the man's force for both wheelbarrow designs, we need to ensure that the wheelbarrow is in equilibrium. This means that the sum of the forces acting on the wheelbarrow must be zero, and the sum of the torques must also be zero.
Let's analyze the torques about the point where the tire contacts the ground. Since the force applied by the man's hands is directed vertically upwards, it does not create any torque relative to this point. Therefore, the only torque acting on the wheelbarrow is due to the weight of the load.
Design 1:
In this design, the load chamber is positioned closer to the wheel, resulting in a shorter lever arm for the load torque. Let's denote the distance from the point where the tire contacts the ground to the center of mass of the load chamber as d1. The lever arm for the weight of the load is then d1.
To maintain equilibrium, the torque created by the load must be balanced by the torque created by the force at the handles. Let's denote the magnitude of the man's force as F1.
The torque due to the load is given by the product of the weight of the load (WL) and the lever arm (d1):
Torque_load = WL * d1
Since the wheelbarrow is in equilibrium, the torque due to the man's force must be equal and opposite:
Torque_man = - F1 * L1
where L1 is the distance from the point where the tire contacts the ground to the point where the man's force is applied vertically upwards.
Since the sum of the torques must be zero, we have:
Torque_load + Torque_man = 0
WL * d1 - F1 * L1 = 0
From this equation, we can solve for F1:
F1 = (WL * d1) / L1
Design 2:
In this design, the load chamber is positioned further away from the wheel, resulting in a longer lever arm for the load torque. Let's denote the distance from the point where the tire contacts the ground to the center of mass of the load chamber as d2. The lever arm for the weight of the load is then d2.
Similar to design 1, let's denote the magnitude of the man's force as F2.
Using the same equilibrium condition, we have:
Torque_load = WL * d2
Torque_man = - F2 * L2
where L2 is the distance from the point where the tire contacts the ground to the point where the man's force is applied vertically upwards.
Again, using the condition that the sum of the torques is zero, we have:
Torque_load + Torque_man = 0
WL * d2 - F2 * L2 = 0
From this equation, we can solve for F2:
F2 = (WL * d2) / L2
By plugging in the given values for WL, d1, d2, L1, and L2, we can calculate the magnitudes of the man's force for both designs.