An object is swinging from a point that is horizontal compared to the lowest point the object will reach (where the rope is vertical). If the length of the rope is 2 meters and the mass of the object is 30 kg, what is the tension in the rope at the bottom of the swing?
Need Step by step solution!
Drop in potential energy from horizontal to vertical orientation is
2 meters *m* g
This must equal 1/2 m v^2, because velocity is zero when the rope is horizontal. This means that:
v^2 = 4 meters*g --->
v^2/(2meters) = 2 g is the centripetal acceleration.
Newton's second law:
F = m a
applied to the object gives:
-mg + T = 2mg
where T is the tension and we take the positive direction to be upward.
The tension is thus T = 3 m g.
Is it vertical vs. Horizontal energy to get the Acceleration?
NVM you are relating GPE for Horizontal to KE of Horizontal.
To find the tension in the rope at the bottom of the swing, you can use the concept of centripetal force. The centripetal force is the force acting towards the center of a circular motion and is given by the equation:
Fc = (m * v^2) / r
where Fc is the centripetal force, m is the mass of the object, v is the velocity of the object, and r is the radius of the circular motion.
In this case, the object is swinging in a vertical circle, so the radius of the circular motion is the length of the rope, which is 2 meters.
To find the velocity, you can use the conservation of mechanical energy. As the object swings from the highest point to the lowest point, it loses potential energy and gains kinetic energy. At the highest point, all the energy is potential energy, and at the lowest point, all the energy is kinetic energy. The equation for conservation of mechanical energy is:
PE + KE = constant,
where PE is the potential energy, KE is the kinetic energy, and the constant value is the total energy of the system.
Given that at the highest point, the potential energy is equal to the maximum potential energy, which is given by m * g * h, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the swing.
At the lowest point, the potential energy is zero, and the kinetic energy is equal to the total energy of the system.
Setting up the equation:
m * g * h = (1/2) * m * v^2.
Simplifying and solving for v:
v^2 = 2 * g * h.
Now, we can calculate the velocity v.
First, we need to find the height h. Since the rope is vertical at the lowest point, the height h is equal to the length of the rope, which is 2 meters.
Now, we can plug in the values:
v^2 = 2 * 9.8 m/s^2 * 2 m,
v^2 = 39.2 m^2/s^2.
After taking the square root, we obtain:
v ≈ 6.26 m/s.
Finally, we can use the centripetal force equation to find the tension in the rope at the bottom of the swing:
Fc = (m * v^2) / r,
Fc = (30 kg * (6.26 m/s)^2) / 2 m,
Fc ≈ 587.1 N.
Therefore, the tension in the rope at the bottom of the swing is approximately 587.1 Newtons.