A rod is lying on the top of a table. One end of the rod is hinged to the table so that the rod can rotate freely on the tabletop. Two forces, both parallel to the tabletop, act on the rod at the same place. One force is directed perpendicular to the rod and has a magnitude of 40.5 N. The second force has a magnitude of 49.5 N and is directed at an angle θ with respect to the rod. If the sum of the torques due to the two forces is zero, what must be the angle θ?
A flywheel is a solid disk that rotates about an axis that is perpendicular to the disk at its center. Rotating flywheels provide a means for storing energy in the form of rotational kinetic energy and are being considered as a possible alternative to batteries in electric cars. The gasoline burned in a 350-mile trip in a typical midsize car produces about 1.90 109 J of energy. How fast would a 18-kg flywheel with a radius of 0.23 m have to rotate to store this much energy? Give your answer in rev/min.
To solve this problem, we need to find the angle θ at which the second force is directed with respect to the rod.
Let's analyze the torques caused by each force separately. Torque is calculated as the product of the force and the perpendicular distance from the point of rotation (hinge) to the line of action of the force.
1. Torque caused by the perpendicular force:
The magnitude of the perpendicular force is 40.5 N. Since it is perpendicular to the rod, the perpendicular distance from the hinge to the line of action of the force is the length of the rod.
2. Torque caused by the angled force:
The magnitude of the angled force is 49.5 N. To calculate the torque caused by this force, we need to find the perpendicular distance from the hinge to the line of action of the force.
Since we are given that the sum of the torques due to both forces is zero, we can set up an equation:
Torque caused by perpendicular force = Torque caused by angled force
Using the equation for torque (τ = r x F, where τ is torque, r is the perpendicular distance, and F is the force), we have:
(L) * (40.5) = (L * sin(θ)) * (49.5)
Here, L represents the length of the rod.
Simplifying the equation, we get:
40.5 = sin(θ) * 49.5
Now, we can solve this equation to find the angle θ.
To solve this problem, we need to apply the concept of torques and equilibrium.
Torque is the measure of the force's ability to cause an object to rotate about a specified axis. In this case, the rotation axis is the hinge point where the rod is attached to the table.
The torque (τ) produced by a force (F) acting at a distance (r) from the axis of rotation can be calculated using the formula:
τ = F * r * sin(θ)
To determine the angle θ, we need to find the net torque acting on the rod. Since the sum of the torques is zero, the torques due to the two forces must cancel each other out.
Let's calculate the torques due to each force:
1. Torque due to the perpendicular force (40.5 N):
Since this force is perpendicular to the rod, it will create a torque of zero because sin(90) = 0.
2. Torque due to the second force (49.5 N):
This force is directed at an angle θ with respect to the rod. The distance from the force to the hinge point is not given. However, since both forces act at the same place, we can assume that the distance is the same for both forces. Let's call this distance "r".
Using the torque formula, the torque due to the second force is:
τ₂ = 49.5 * r * sin(θ)
Since the total torque is zero, we can write the equation:
τ₁ + τ₂ = 0
Since τ₁ = 0, we can simplify the equation to:
τ₂ = 0
Substituting the expression for τ₂, we have:
49.5 * r * sin(θ) = 0
For the equation to be true, either the magnitude of the force (49.5 N) or the sine of the angle (sin(θ)) has to be zero.
Considering the force magnitude cannot be zero, we have:
sin(θ) = 0
To find the angle θ, we need to find the values of θ where the sine of θ equals zero.
In trigonometry, the sine of an angle is zero at θ = 0, 180°, 360°, etc. These angles are known as the zero crossings of the sine function.
Thus, the possible values for the angle θ are:
θ = 0°, 180°, 360°, etc.
Therefore, the angle θ can be any multiple of 180°.