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Mobile App Chapter 11: Problem 35
A construction supervisor of mass \(M=92.1 \mathrm{~kg}\) is standing on a board of mass \(m=27.5 \mathrm{~kg} .\) Two sawhorses at a distance \(\ell=3.70 \mathrm{~m}\) apart support the board. If the man stands a distance \(x_{1}=1.07 \mathrm{~m}\) away from the left-hand sawhorse as shown in the figure, what is the force that the board exerts on that sawhorse?
Short Answer
Expert verified
Answer: The force exerted by the board on the left-hand sawhorse is 839.079 N.
Step by step solution
01
Identify the forces acting on the system
The forces acting on the system are the gravitational force of the construction supervisor, the gravitational force of the board, and the forces exerted by the left and right sawhorses on the board. We'll need to calculate the magnitudes of these forces to analyze the torques at play.
02
Calculate gravitational forces
Let's calculate the gravitational forces acting on the supervisor and the board.
\(F_{M} = M \cdot g\) (gravitational force exerted on the supervisor)
\(F_{m} = m \cdot g\) (gravitational force exerted on the board)
where \(g = 9.81 \mathrm{m/s^2}\) is the acceleration due to gravity.
\(F_{M} = 92.1 \times 9.81 = 903.231 \mathrm{N}\)
\(F_{m} = 27.5 \times 9.81 = 269.775 \mathrm{N}\)
03
Apply Torque equilibrium equation
In order for the system to be in equilibrium, the sum of the torques about an arbitrary point must be equal to zero. Let the distance x2 be the distance from the construction supervisor to the right-hand sawhorse. Then,
\(x_{2} = \ell - x_{1}\)
Let's consider the torques about the left-hand sawhorse. The forces that contribute a torque about the left-hand sawhorse are: \(F_{M}\), \(F_{m}\), and the force exerted by the right-hand sawhorse (we'll call it \(F_{R}\)). The force exerted by the left-hand sawhorse does not contribute to the torque because its line of action passes through the axis of rotation. Therefore,
\(T_{M} = F_{M} \cdot x_{1}\) (torque due to the supervisor)
\(T_{m} = F_{m} \cdot \frac{\ell}{2}\) (torque due to the board)
\(T_{R} = F_{R} \cdot \ell\) (torque due to the right-hand sawhorse)
The torque equilibrium equation is:
\(T_{M} + T_{m} - T_{R} = 0\)
04
Solve for the force exerted by the left-hand sawhorse
Using the torque equilibrium equation derived in Step 3, let's calculate the force exerted by the right-hand sawhorse, \(F_{R}\).
\(F_{R} = \frac{T_{M} + T_{m}}{\ell}\)
\(F_{R} = \frac{(903.231 \times 1.07) + (269.775 \times \frac{3.70}{2})}{3.70} = 333.927 \mathrm{N}\)
Now, we'll find the force exerted by the left-hand sawhorse. The sum of the forces in the vertical direction should be equal to zero since the system is in equilibrium:
\(F_{L} - F_{M} - F_{m} + F_{R} = 0\)
Solving for \(F_{L}\) (force exerted by the left-hand sawhorse):
\(F_{L} = F_{M} + F_{m} - F_{R}\)
\(F_{L} = 903.231 + 269.775 - 333.927 = 839.079 \mathrm{N}\)
So the force that the board exerts on the left-hand sawhorse is \(839.079 \mathrm{N}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Torque Equilibrium Equation
Understanding the concept of torque is essential in analyzing situations involving rotational forces. Torque, also called moment of force, measures the tendency of a force to rotate an object about an axis or pivot.
In physics, when we say a system is in 'static equilibrium', we mean the following two conditions must be met: firstly, the sum of all forces acting upon the system is zero, ensuring there is no net linear acceleration. Secondly, the sum of all torques about any axis is also zero, preventing any rotational acceleration.
The torque equilibrium equation is a mathematical expression of the latter condition. It states that for an object to be in a state of static rotational equilibrium, the sum of clockwise torques must equal the sum of counterclockwise torques. This can be represented as:\[ \sum \tau_{\text{clockwise}} = \sum \tau_{\text{counterclockwise}} \]
In simpler terms, for any fulcrum or pivot point you choose, the torques that would cause rotation in one direction are perfectly balanced by the torques that would cause rotation in the opposite direction. In the context of the construction supervisor standing on the board, we consider torques about one of the sawhorses and ensure that the torques created by the gravitational forces of both the supervisor and the board, as well as the reaction force from the other sawhorse, balance out.
When solving problems that involve torque and static equilibrium, it is helpful to draw a free body diagram and label all the forces and distances involved. This makes it simpler to apply the torque equilibrium equation correctly.
In physics, when we say a system is in 'static equilibrium', we mean the following two conditions must be met: firstly, the sum of all forces acting upon the system is zero, ensuring there is no net linear acceleration. Secondly, the sum of all torques about any axis is also zero, preventing any rotational acceleration.
The torque equilibrium equation is a mathematical expression of the latter condition. It states that for an object to be in a state of static rotational equilibrium, the sum of clockwise torques must equal the sum of counterclockwise torques. This can be represented as:\[ \sum \tau_{\text{clockwise}} = \sum \tau_{\text{counterclockwise}} \]
In simpler terms, for any fulcrum or pivot point you choose, the torques that would cause rotation in one direction are perfectly balanced by the torques that would cause rotation in the opposite direction. In the context of the construction supervisor standing on the board, we consider torques about one of the sawhorses and ensure that the torques created by the gravitational forces of both the supervisor and the board, as well as the reaction force from the other sawhorse, balance out.
When solving problems that involve torque and static equilibrium, it is helpful to draw a free body diagram and label all the forces and distances involved. This makes it simpler to apply the torque equilibrium equation correctly.
Gravitational Force Calculation
The force of gravity, which we often refer to as weight in everyday terms, plays a significant role in problems involving static equilibrium. To calculate the gravitational force acting on an object, we use the following formula:\[ F_{\text{gravity}} = m \cdot g \]
where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity, which is approximately \( 9.81 \text{ m/s}^2 \) on Earth's surface. For the construction supervisor and the board in our problem, we calculate two separate gravitational forces, because each has a different mass.
Understanding how to calculate these forces is crucial for determining the forces exerted by the supports, since they must counteract the gravitational force to keep the system in equilibrium. This means that the stronger the gravity or the greater the mass, the more force the supports must exert. In the real world, accurate gravitational force calculations are key to constructing safe and stable structures.
where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity, which is approximately \( 9.81 \text{ m/s}^2 \) on Earth's surface. For the construction supervisor and the board in our problem, we calculate two separate gravitational forces, because each has a different mass.
Understanding how to calculate these forces is crucial for determining the forces exerted by the supports, since they must counteract the gravitational force to keep the system in equilibrium. This means that the stronger the gravity or the greater the mass, the more force the supports must exert. In the real world, accurate gravitational force calculations are key to constructing safe and stable structures.
Equilibrium of Forces
The equilibrium of forces is a foundational concept in both statics and dynamics. It refers to a state where all forces acting on a body balance each other out, such that there is no net force causing the body to accelerate. For equilibrium to be maintained, the following condition must hold:\[ \sum \vec{F} = 0 \]
This vector sum of forces equals zero indicates that the forces in any given direction are counterbalanced by forces of the same magnitude in the opposite direction.
In the case of the construction supervisor and the board, the equilibrium of forces is achieved when the vertical forces due to the board's and supervisor's weights are exactly counteracted by the upward forces exerted by the sawhorses. This explains why one sawhorse must exert a force equal to the combined weight of the supervisor and the board, minus the force exerted by the other sawhorse.
To ensure students grasp this concept, it's essential to emphasize that in problems like these, the role of gravity is countered by the normal force exerted by the surfaces or supports in contact with the object. By meticulously calculating each force and ensuring that the net force is null, students can solve for unknown forces, such as those in the sawhorses, and understand the conditions for static equilibrium.
This vector sum of forces equals zero indicates that the forces in any given direction are counterbalanced by forces of the same magnitude in the opposite direction.
In the case of the construction supervisor and the board, the equilibrium of forces is achieved when the vertical forces due to the board's and supervisor's weights are exactly counteracted by the upward forces exerted by the sawhorses. This explains why one sawhorse must exert a force equal to the combined weight of the supervisor and the board, minus the force exerted by the other sawhorse.
To ensure students grasp this concept, it's essential to emphasize that in problems like these, the role of gravity is countered by the normal force exerted by the surfaces or supports in contact with the object. By meticulously calculating each force and ensuring that the net force is null, students can solve for unknown forces, such as those in the sawhorses, and understand the conditions for static equilibrium.
