8) A rock of mass 400 g is tied to one end of a string that is 2.0 m in length and swung around in a horizontal circle. It takes 2 s to complete one circle.
a. Find the angular velocity of the rock just as the string breaks.
b. At what speed is the rock traveling just as the string breaks?
c. What is the centripetal force swinging the rock.
a=3.14 rads
To solve the given problems, we need to use the following formulas related to circular motion:
1. Angular velocity (ω) = 2πf
2. Centripetal acceleration (a) = ω²r
3. Centripetal force (F) = ma_c = m(ω²r)
Now, let's solve the problems step by step:
a. Find the angular velocity of the rock just as the string breaks.
To find the angular velocity, we need the frequency (f). The frequency is the reciprocal of the time taken to complete one circle (T).
Given:
- Time taken to complete one circle (T) = 2 s
We can calculate the frequency using the formula:
f = 1 / T
Substituting the value:
f = 1 / 2
Now, we can calculate the angular velocity (ω) using the formula:
ω = 2πf
Substituting the value:
ω = 2π * (1/2)
ω = π radians/s
Therefore, the angular velocity of the rock just as the string breaks is π radians/s.
b. At what speed is the rock traveling just as the string breaks?
To find the speed at which the rock is traveling, we need to determine the velocity (v). Velocity is the product of the angular velocity (ω) and the radius of the circular path (r).
Given:
- Length of the string (radius), r = 2.0 m
- Angular velocity, ω = π radians/s (calculated in part a)
We can calculate the speed (v) using the formula:
v = ωr
Substituting the known values:
v = π * 2.0
v = 2π m/s
Therefore, the speed at which the rock is traveling just as the string breaks is 2π m/s.
c. What is the centripetal force swinging the rock?
To find the centripetal force, we need the mass of the rock (m) and the acceleration (a). The centripetal force (F) is the product of mass and centripetal acceleration.
Given:
- Mass of the rock, m = 400 g = 0.4 kg (convert grams to kilograms)
Recall the formula for centripetal acceleration:
a = ω²r
Substituting the known values:
a = (π radians/s)² * 2.0 m
a = π² * 2.0 m
Now, we can calculate the centripetal force using the formula:
F = ma
Substituting the value of mass (m) and acceleration (a):
F = 0.4 kg * (π² * 2.0 m)
Calculating the value:
F ≈ 2.51 N (rounded to two decimal places)
Therefore, the centripetal force swinging the rock just as the string breaks is approximately 2.51 N.