Fig. 6-48 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 0.038 kg, the string has length L = 1.2 m and negligible mass, and the bob follows a circular path of circumference 0.75 m. What are (a) the tension in the string and (b) the period of the motion?
To solve this problem, we can use the principles of circular motion. Let's break it down step by step:
Step 1: Calculate the velocity of the bob.
The circumference of the circular path is given as 0.75 m. So, the distance traveled by the bob in one complete revolution is equal to the circumference, which can be calculated using the formula:
C = 2πr
Here, r represents the radius of the circular path. Since the radius is not given directly, we need to find it using the length of the string, L.
The length of the string (L) forms the slant height of the cone (which is the hypotenuse of a right-angled triangle) and the radius (r) of the circular path forms the base of the triangle. The vertical height (h) of the triangle can be calculated using the Pythagorean theorem:
h² = L² - r²
Rearranging the equation, we find:
r = √(L² - h²)
Since the bob moves in a circular path, the vertical height (h) is given by:
h = L - R
Where R represents the height of the cone formed by the string.
Now, substituting the known values:
r = √(1.2² - (1.2 - R)²)
r = √(1.44 - 1.44 + 2.4R - R²)
r = √(2.4R - R²)
The circumference is given as 0.75 m, so:
0.75 = 2πr
Substituting the value of r:
0.75 = 2π√(2.4R - R²)
Now, we can solve this equation to find the value of R, which is the height of the cone.
Step 2: Calculate the tension in the string.
The force of tension in the string provides the necessary centripetal force to keep the bob moving in a circular path. At any point on the circular path, the tension can be calculated using the formula:
T = (m × v²) / r
Here, m represents the mass of the bob (0.038 kg), v represents the velocity of the bob, and r represents the radius of the circular path.
Step 3: Calculate the period of the motion.
The period of the circular motion is the time taken for the bob to complete one full revolution. It can be calculated using the formula:
T = (2πr) / v
We now have all the information required to solve for (a) the tension in the string and (b) the period of the motion.
To solve this problem, we need to break it down into two parts:
(a) Finding the tension in the string:
To determine the tension in the string, we can analyze the forces acting on the bob at the bottom of the pendulum's path. There are two forces acting on the bob: gravitational force (mg) and the tension in the string (T).
Since the bob is moving in a horizontal circle at constant speed, the net force acting on it must be directed towards the center of the circle. In this case, the centripetal force is provided solely by the tension in the string.
Therefore, we can set up the following equation:
T = mv^2 / r
Here, m represents the mass of the bob (0.038 kg), v is the velocity of the bob (which we can find using the circumference and period), and r is the radius of the circular path (0.75 m / 2 = 0.375 m).
(b) Finding the period of the motion:
We know that the circumference of a circle is given by the formula C = 2πr. We are given the circumference (0.75 m) and can rearrange the formula to solve for the radius (r) of the circular path.
r = C / (2π)
= 0.75 m / (2π)
To find the period (T) of the motion, we can use the formula:
T = 2πr / v
Here, r is the radius of the circular path, and v is the velocity of the bob. We can rearrange this formula to solve for the period (T):
T = 2πr / v
= 2π(0.375 m) / v
By substituting the values and solving the equation, we can find the period (T) of the motion.
Using these two equations, we can now find the tension in the string (T) and the period (T) of the motion.
Thanks!
The angle of the pendulum from vertical is
A = arcsin r/L = arcsin [0.75/(2 pi L)]
= 5.71 degrees
You will need this angle later.
The vertical and horizontanl equations of motion are:
T sin A = m V^2/R
T cos A = m g
which tells you that
tan A = 0.100 = V^2/gR
V^2 = 0.100*9.8m/s^2*0.1194m = 0.1170 m^2/s^2
V = 0.342 m/s
(a) T = mg/cosA = 0.038kg*9.8m/s^2/.9950
= 0.3743 Newtons
(b) Period = (circumference)/V
= 2 pi R/sqrt(g R tan A)
for small angles A, tan A = R/L, so
Period = 2 pi R /sqrt (g R *R/L)
= 2 pi sqrt (L/g)
which is the same as the formula for a pendulum oscillating in a plane (one dimension)
Period = 2.2 seconds
Check my work