A 500 N weight is attached to a stiff rod of negligible weight that is hinged to one end. The rod is released from rest in horizontal position and allowed to swing freely in vertical arc. Through what angle with the horizontal must it swing to cause a tension in it of 1000N?
=41.8 degree
To find the angle through which the rod must swing in order to cause a tension of 1000N, we can make use of the principle of conservation of energy.
The potential energy of the weight at the highest point of the swing is equal to the work done by the tension force in lifting the weight to that point. The potential energy is given by the formula:
Potential Energy = mgh
Where:
m = mass of the weight (500 N / g, since weight = mass x gravity)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height to which the weight is lifted
Since the weight is attached to a stiff rod, the height to which the weight is lifted is equal to the radius of the circular path it follows during the swing. Therefore, h = r.
Now, let's denote the angle through which the rod swings from the horizontal position as θ. The radius of the circular path is given by:
r = 2R sin(θ/2)
Where:
R = length of the rod
The work done by the tension force can be calculated as the product of the tension force and the distance over which it acts. The distance over which it acts is given by:
s = 2R (1 - cos(θ/2))
Where:
s = distance
Work = Tension x Distance = Tension x 2R (1 - cos(θ/2))
Since the potential energy at the highest point is equal to the work done by the tension force, we have:
mgh = Tension x 2R (1 - cos(θ/2))
Substituting in the values:
(500 N / g) x g x 2R = 1000 N x 2R (1 - cos(θ/2))
Simplifying:
1000 N = 1000 N (1 - cos(θ/2))
Dividing both sides by 1000 N:
1 = 1 - cos(θ/2)
Rearranging the equation:
cos(θ/2) = 0
Taking the inverse cosine (cos^-1) of both sides:
θ/2 = 90 degrees
Multiplying both sides by 2:
θ = 180 degrees
Therefore, the rod must swing through an angle of 180 degrees with the horizontal to cause a tension of 1000N.
To determine the angle through which the rod must swing in order to cause a tension of 1000 N, we can use the principles of conservation of energy and equilibrium.
First, let's consider the initial state when the rod is horizontal. At this point, the weight is only acting vertically downward and there is no tension in the rod. The gravitational potential energy is also at its minimum.
Now, as the rod swings downward, its potential energy decreases while the tension in the rod increases. At the lowest point of the swing, all of the potential energy is converted into kinetic energy, and the tension in the rod is at its maximum.
At the highest point of the swing, the tension in the rod is again at its minimum, while the potential energy is at its maximum. However, since we are interested in the angle where the tension is 1000 N, we need to find the position where the tension is maximum.
To do this, we can set up an equation using the principle of equilibrium. At the lowest point of the swing, the forces acting on the weight are the tension in the rod (T), the weight (mg), and the centripetal force (mv²/r, where v is the velocity and r is the radius of the arc).
Since the rod is released from rest, the initial velocity is zero. Therefore, the tension at the lowest point of the swing is equal to the sum of the weight and centripetal force:
T = mg + mv²/r
Given that the weight (mg) is 500 N and the tension (T) is 1000 N, we can substitute these values into the equation:
1000 N = 500 N + (0/9.8 m/s²) * v²/r
Simplifying, we get:
500 N = (v²/r) * (1/9.8 m/s²)
Solving for v²/r, we have:
(v²/r) = (500 N) * (9.8 m/s²)
Now, we need one more equation to relate the radius (r) and the angle (θ). The length of the rod, when it is at an angle θ, is given by the equation:
s = rθ
where s is the arc length. In this case, s is equal to the length of the rod.
Since the rod is stiff and negligible in weight, the length remains constant. Therefore, s is the same at any angle. We can rearrange the equation to solve for r:
r = s/θ
Now we can substitute this equation into our previous equation:
(v²/(s/θ)) = (500 N) * (9.8 m/s²)
v² = (500 N) * (9.8 m/s²) * (s/θ)
Finally, we know that the velocity (v) at the lowest point is equal to the velocity at the highest point. So, we can rewrite the equation as:
(v²/2) = (500 N) * (9.8 m/s²) * (s/θ)
To simplify, let's assume that the length of the rod (s) is 1 meter:
(v²/2) = (500 N) * (9.8 m/s²) * (1 meter/θ)
Now, we have an equation that relates the final angle (θ) with the tension in the rod (1000 N). We can solve this equation to find the value of θ.