A woman holds a 1.9 m long uniform 5.5 kg pole as shown in the figure .
her hands are 30 cm apart.
Determine the magnitudes of the forces she must exert with each hand.
To what position should she move her left hand so that neither hand has to exert a force greater than 110 N?
To what position should she move her left hand so that neither hand has to exert a force greater than 35 N?
To determine the magnitudes of the forces she must exert with each hand, we need to analyze the rotational equilibrium of the pole. We can use the condition that the sum of the torques acting on the pole is zero.
Let's denote the distance between the left hand and the center of gravity of the pole as x. The distance between the right hand and the center of gravity is then (1.9 m - x). The center of gravity of the pole can be considered as the midpoint of the pole.
First, we need to calculate the position of the center of gravity. As the pole is uniform, the center of gravity is located at the midpoint, which is given by (1.9 m / 2) = 0.95 m.
Since the pole is in equilibrium, the sum of the torques acting on it must be zero. The torque exerted by the left hand can be calculated as follows:
TorqueL = ForceL * DistanceL
Where ForceL is the force exerted by the left hand and DistanceL is the distance between the left hand and the center of gravity.
Similarly, the torque exerted by the right hand can be calculated as:
TorqueR = ForceR * DistanceR
Where ForceR is the force exerted by the right hand and DistanceR is the distance between the right hand and the center of gravity.
Since the sum of the torques is zero, we have:
TorqueL + TorqueR = 0
ForceL * DistanceL + ForceR * DistanceR = 0
Substituting the values, we have:
ForceL * x + ForceR * (1.9 m - x) = 0
Now, we can solve this equation for the magnitudes of the forces.
To find the position to move the left hand so that neither hand has to exert a force greater than 110 N, we can set the maximum force equal to 110 N.
110 N * x + ForceR * (1.9 m - x) = 0
Simplifying, we can solve this equation for x, the position of the left hand.
To find the position to move the left hand so that neither hand has to exert a force greater than 35 N, we can set the maximum force equal to 35 N.
35 N * x + ForceR * (1.9 m - x) = 0
Simplifying, we can solve this equation for x, the position of the left hand.
By solving these equations, we can find the positions for the left hand in order to keep the forces exerted by both hands below a certain value.