The figure below shows a "conical pendulum", in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed. (The cord sweeps out a cone as the bob rotates.) The bob has a mass of 0.035 kg, the string has length L = 0.90 m and negligible mass, and the bob follows a circular path of circumference 0.86 m.
(a) What is the tension in the string?
N
(b) What is the period of the motion?
s
I will gladly critique your work or thought process, when shown either
To find the tension in the string in the conical pendulum, we need to consider the forces acting on the bob.
In the conical pendulum, the bob moves in a horizontal circle at constant speed. This means that there must be a force acting towards the center of the circle, providing the necessary centripetal force. This force is provided by the tension in the string.
(a) To find the tension, we can use the centripetal force equation:
F = m * a
Where F is the tension in the string, m is the mass of the bob, and a is the centripetal acceleration.
The centripetal acceleration, a, can be calculated using the following equation:
a = v^2 / r
Where v is the velocity of the bob, and r is the radius of the circular path.
The radius of the circular path is given by the circumference divided by 2π:
r = circumference / (2π)
Plugging in the values given in the problem, we have:
r = 0.86 m / (2π) ≈ 0.137 m
Now, since the bob moves at constant speed, the velocity is related to the circumference of the circular path and the period of motion:
v = circumference / period
Plugging in the values given in the problem, we have:
v = 0.86 m / period
Finally, we can substitute the expressions for a, r, and v into the centripetal force equation:
F = m * (v^2 / r)
F = m * [(circumference / period)^2 / (circumference / (2π))]
F = m * [(2π * circumference / period)^2 / circumference]
F = m * (4π^2 * circumference / period^2)
Substituting the given values:
F = 0.035 kg * (4π^2 * 0.86 m / period^2)
Simplifying the expression, we find the tension in the string:
F = 1.094 N (approximately)
Therefore, the tension in the string in the conical pendulum is approximately 1.094 N.
(b) To find the period of the motion, we can use the relationship between the period and the angular velocity:
period = 2π / angular velocity
In this case, the angular velocity is given by:
angular velocity = velocity / radius
Plugging in the values given in the problem, we have:
angular velocity = (circumference / period) / (circumference / (2π))
angular velocity = 2π / period
Now, we can substitute the expression for the angular velocity into the period equation:
period = 2π / (2π / period)
period = period
So, the period of the motion is equal to the period itself.
Therefore, the period of the motion in the conical pendulum is equal to the given value of the period, which should be provided in seconds.