A stationary 3.0-m board of mass 6.0 kg is hinged at one end. A force is applied vertically at the other end, and the board makes a 30° angle with the horizontal. A 40-kg block rests on the board 80 cm from the hinge as shown in the figure below.
a) Find the magnitude of the force F
b) Find the force exerted by the hinge.
c) Find the magnitude of the force Farrowbolditalic as well as the force exerted by the hinge (FH), if F is exerted, instead, at right angles to the board.
some shizzzz
To find the magnitude of the force F in part (a), we can use the principle of torque equilibrium. The torque equation provides a relationship between the forces acting on an object and the distance from the point of rotation (or pivot). Mathematically, torque is defined as the product of the applied force and the perpendicular distance from the point of rotation to the line of action of the force.
a) The torque equation for this situation can be written as:
Torque_due_to_weight = Torque_due_to_Force_F
Torque_due_to_weight = mass_of_block * gravitational_acceleration * distance_from_hinge_to_block * sin(angle_with_horizontal)
Torque_due_to_Force_F = Force_F * distance_from_hinge_to_F * sin(90° - angle_with_horizontal)
Since the board is in equilibrium, the torque due to the weight of the block must be equal to the torque due to Force F. Substituting the given values:
mass_of_block = 40 kg
gravitational_acceleration = 9.8 m/s^2
distance_from_hinge_to_block = 80 cm = 0.8 m
angle_with_horizontal = 30°
We can calculate the torque due to weight:
Torque_due_to_weight = (40 kg)(9.8 m/s^2)(0.8 m)(sin 30°)
Next, we can calculate the distance from the hinge to Force F:
distance_from_hinge_to_F = 3.0 m - 0.8 m = 2.2 m
Now we can set up and solve the equation for torque equilibrium:
Torque_due_to_weight = Torque_due_to_Force_F
(40 kg)(9.8 m/s^2)(0.8 m)(sin 30°) = Force_F(2.2 m)(sin 60°)
Simplifying the equation and solving for Force F:
Force_F = [(40 kg)(9.8 m/s^2)(0.8 m)(sin 30°)] / [(2.2 m)(sin 60°)]
Calculate the value on the right side of the equation to find the magnitude of Force F.
b) To find the force exerted by the hinge, we can use Newton's second law, which states that the sum of the forces acting on an object is equal to its mass times acceleration (F = ma). Since the board is at rest (stationary), the net force acting on it must be zero. Therefore, we can calculate the force exerted by the hinge by considering the forces in the vertical direction.
Force_upwards - Force_due_to_Weight_of_Board - Force_due_to_Weight_of_Block = 0
Force_exerted_by_hinge - (mass_of_board)(gravitational_acceleration) - (mass_of_block)(gravitational_acceleration) = 0
Solve the equation for Force_exerted_by_hinge to find the magnitude of the force exerted by the hinge.
c) If the force F is exerted at right angles to the board (perpendicular to the board), the angle with the horizontal would be 90 degrees. In this case, to find the magnitude of force F and the force exerted by the hinge, we can use the same torque equilibrium equation as in part (a), but with the new angle.
Torque_due_to_weight = Torque_due_to_Force_F
Torque_due_to_weight = (mass_of_block)(gravitational_acceleration)(distance_from_hinge_to_block)(sin 90°)
Torque_due_to_Force_F = Force_F(2.2 m)(sin 0°)
Simplify the equation and solve for Force F to find the magnitude of force F. Then use Newton's second law to find the magnitude of the force exerted by the hinge, as explained in part (b).