Instead of moving back and forth, a conical pendulum moves in a circle at constant speed as its string traces out a cone (see figure below). One such pendulum is constructed with a string of length L = 14.9 cm and bob of mass 0.326 kg. The string makes an angle 𝜃 = 6.84° with the vertical.
(a) What is the radial acceleration of the bob?
[answer] magnitude m/s^2
(b) What is the horizontal and vertical components of the tension force exerted by the string on the bob? (Assume radially inward to be the positive x axis and vertically upward to be the positive y axis. Express your answer in vector form.)
To find the radial acceleration of the bob in the conical pendulum, we can use the following formula:
Radial acceleration (ar) = (velocity (v))^2 / radius (r)
To find the velocity, we can use the fact that the pendulum moves in a circle at a constant speed. This means that the magnitude of the velocity is constant.
To find the magnitude of the velocity, we can use trigonometry. The vertical component of the velocity is determined by the component of gravitational force acting along the string, which can be calculated as:
Vertical component of velocity (v_y) = √(2 * g * L * (1 - cos(θ)))
Where g is the acceleration due to gravity (approximately 9.8 m/s^2), L is the length of the string (14.9 cm or 0.149 m), and θ is the angle the string makes with the vertical (6.84°).
Similarly, the horizontal component of the velocity is determined by the component of gravitational force perpendicular to the string, which can be calculated as:
Horizontal component of velocity (v_x) = √(2 * g * L * sin(θ))
Now that we have the magnitude of the velocity, we can calculate the radial acceleration. The radius (r) is the length of the string (L).
Radial acceleration (ar) = (v^2) / r
Substituting the values we have:
Radial acceleration (ar) = ((v_x^2) + (v_y^2)) / L
Finally, to find the horizontal and vertical components of the tension force exerted by the string on the bob, we can use the following formulas:
Horizontal component of tension force (T_x) = m * ar * sin(θ)
Vertical component of tension force (T_y) = m * ar * cos(θ)
Where m is the mass of the bob (0.326 kg), ar is the radial acceleration, and θ is the angle the string makes with the vertical.
To find the answers for both parts (a) and (b), we can use the following formulas:
(a) The radial acceleration of the bob in a conical pendulum is given by the formula:
a_r = ω^2 * r
where a_r is the radial acceleration, ω is the angular velocity, and r is the radius of the circular motion.
(b) The tension force exerted by the string on the bob can be split into horizontal and vertical components. The horizontal component of the tension force is equal to the centripetal force, and the vertical component of the tension force is equal to the gravitational force acting on the bob.
Now, let's calculate the answers step-by-step.
(a) Calculating the radial acceleration:
- Firstly, convert the given angle from degrees to radians:
θ = 6.84° * (π/180) = 0.1193 rad
- Next, calculate the radius of the circular motion:
r = L * sin(θ)
= 0.149 m * sin(0.1193)
≈ 0.0294323 m
- Finally, calculate the angular velocity:
ω = √(g / r)
= √(9.8 m/s^2 / 0.0294323 m)
≈ 17.9086 rad/s
Now we can calculate the radial acceleration:
a_r = ω^2 * r
= (17.9086 rad/s)^2 * 0.0294323 m
≈ 9.4518 m/s^2
The magnitude of the radial acceleration is approximately 9.4518 m/s^2.
(b) Calculating the horizontal and vertical components of the tension force:
- The horizontal component of the tension force is equal to the centripetal force, which can be calculated using the formula:
F_c = m * a_r
- The vertical component of the tension force is equal to the gravitational force acting on the bob, which can be calculated using the formula:
F_g = m * g
where m is the mass of the bob and g is the acceleration due to gravity.
Now let's calculate the tension force components:
Using the given mass:
m = 0.326 kg
Using the acceleration due to gravity:
g = 9.8 m/s^2
The horizontal component of the tension force (T_x):
T_x = F_c = m * a_r
= 0.326 kg * 9.4518 m/s^2
≈ 3.081 N
The vertical component of the tension force (T_y):
T_y = F_g = m * g
= 0.326 kg * 9.8 m/s^2
≈ 3.2048 N
So, the horizontal component of the tension force is approximately 3.081 N, and the vertical component of the tension force is approximately 3.2048 N.
Therefore, the answer to part (a) is 9.4518 m/s^2, and the answer to part (b) is T = (3.081 N, 3.2048 N) in vector form.