A small sphere of mass 2 kg is attached to the end of a cord of length 2m and rotated in a vertical circle about a fixed point O. Determine the tension in the cord at the lowest point of the trajectory if the speed at this point is 7m/s. Use g=9.8 m/s2.
Well, I hope that sphere shaved its legs before entering this vertical circle! Let's solve this gravity-defying riddle together.
Firstly, we need to find out the acceleration of the sphere at the lowest point of its trajectory. We can do this by using the centripetal acceleration formula:
a = (v^2) / r
where v is the velocity and r is the radius of the circle. In this case, the velocity is given as 7 m/s and the radius is the length of the cord, which is 2m. Plugging in these values, we have:
a = (7^2) / 2 = 49 / 2 = 24.5 m/s^2
Since the sphere is at the lowest point of its trajectory, the net force acting on it is equal to the tension in the cord plus its weight. At this point, the weight is given by:
weight = m * g
where m is the mass of the sphere (2 kg) and g is the gravitational acceleration (9.8 m/s^2). So, the weight of the sphere is:
weight = 2 * 9.8 = 19.6 N
Now, the net force can be written as:
net force = tension - weight
Since the sphere is moving in a circle, the net force is equal to the centripetal force:
net force = m * a
where a is the centripetal acceleration we calculated earlier. Plugging in the values, we have:
m * a = tension - weight
2 * 24.5 = tension - 19.6
49 = tension - 19.6
tension = 49 + 19.6
tension = 68.6 N
So, the tension in the cord at the lowest point of the trajectory is approximately 68.6 N.
Just remember, this tension business is no laughing matter for the poor sphere. Take care of those cords!
To find the tension in the cord at the lowest point of the trajectory, we can analyze the forces acting on the small sphere at that point.
At the lowest point, the tension in the cord provides the centripetal force necessary to keep the sphere moving in a circle. The gravitational force acting on the sphere can be divided into two components: the component acting along the vertical direction (mg), and the component acting along the cord (mgcosθ), where θ is the angle between the cord and the vertical direction.
Since the sphere is moving in a vertical circle, at the lowest point, the gravitational force is directed upward and the tension in the cord is directed downward. The equation for the net force at the lowest point is:
Tension - mgcosθ = m * centripetal acceleration
The centripetal acceleration is given by the formula:
Centripetal acceleration = (v^2)/r
Where v is the speed at the lowest point and r is the radius of the circle, which is equal to the length of the cord.
Plugging in the given values:
v = 7 m/s (speed at the lowest point)
r = 2 m (length of the cord)
g = 9.8 m/s^2 (acceleration due to gravity)
The equation becomes:
Tension - mgcosθ = m * (v^2/r)
Substituting the values into the equation:
Tension - (2 kg * 9.8 m/s^2 * cos(θ)) = 2 kg * (7 m/s)^2 / 2 m
Simplifying:
Tension - 19.6 N * cos(θ) = 49 N
To find the value of cos(θ), we can use the fact that the tension in the cord is maximum at the topmost point of the trajectory, where the sphere is momentarily at rest. Therefore, at the topmost point, the centripetal force is provided by only the tension in the cord:
Tension = mg + mgcosθ
Substituting the values:
Tension = 2 kg * 9.8 m/s^2 + 2 kg * 9.8 m/s^2 * cos(θ)
Simplifying:
Tension = 19.6 N + 19.6 N * cos(θ)
At the topmost point, the speed is zero (v=0), so the centripetal acceleration is also zero. The equation becomes:
Tension = 19.6 N + 0
Tension = 19.6 N
Therefore, at the topmost point, the tension in the cord is 19.6 N.
Now we can substitute this value back into the equation we obtained earlier:
19.6 N - 19.6 N * cos(θ) = 49 N
Simplifying:
-19.6 N * cos(θ) = 49 N - 19.6 N
-19.6 N * cos(θ) = 29.4 N
Dividing both sides by -19.6 N:
cos(θ) = -29.4 N / -19.6 N
cos(θ) ≈ 1.5
Since the tension in the cord is directed downward at the lowest point, cos(θ) will be negative. However, cos(θ) cannot be larger than 1. Therefore, there is an error in the problem statement or the calculations.
Please re-check the given information or provide additional details to help resolve the discrepancy.
To determine the tension in the cord at the lowest point of the trajectory, we can start by considering the forces acting on the sphere at that point.
At the lowest point, the sphere is moving in a circular path, so its acceleration is directed towards the center of the circle. The net force acting on the sphere is the sum of the tension force (T) in the cord and the force of gravity (mg).
Since the sphere is moving in a circle, we can use centripetal force to write an equation:
T - mg = m * (v^2 / r)
Here:
T is the tension in the cord,
m is the mass of the sphere (2 kg),
g is the acceleration due to gravity (9.8 m/s^2),
v is the speed at the lowest point (7 m/s),
and r is the radius of the circle (equal to the length of the cord, 2m).
We can substitute the given values into the equation:
T - (2 kg)(9.8 m/s^2) = (2 kg) * (7 m/s)^2 / 2 m
Simplifying that equation gives:
T - 19.6 N = 49 N
Now, we isolate the Tension T:
T = 49 N + 19.6 N
T = 68.6 N
Therefore, the tension in the cord at the lowest point of the trajectory is 68.6 Newtons.