A boy ties a stone to the end of a spring which he then whirles above his head round a circular path of radius 15 cm if the stone makes 20 oscillations in 10 secs. Calculate the angular and linear speeds of the stone
2 oscillations per second ... 4 π rad/s
radius is 15 cm ... 4 * π * 15 cm/s
Why did the stone become a fashion influencer? Because it was always spinning in style! Now, let's calculate the angular and linear speeds of this fashionable stone.
First, let's find the time period of one oscillation. Since the boy made 20 oscillations in 10 seconds, the time period of one oscillation is 10 seconds divided by 20 oscillations, which gives us 0.5 seconds.
Now, the angular speed can be calculated using the formula:
Angular Speed (ω) = 2π / Time Period
Plugging in the values, we get:
Angular Speed (ω) = 2π / 0.5 s
Angular Speed (ω) = 4π rad/s
The linear speed can be found using the formula:
Linear Speed (v) = Angular Speed (ω) * Radius
Substituting the values, we have:
Linear Speed (v) = 4π rad/s * 0.15 m
Linear Speed (v) ≈ 1.88 m/s
So, the angular speed of the stone is 4π rad/s and the linear speed is approximately 1.88 m/s. That stone is definitely rocking those speeds!
To calculate the angular speed of the stone, we need to find the total angle covered by the stone in one second.
The stone completes 20 oscillations in 10 seconds, so in one second it completes 2 oscillations.
The circumference of the circular path is calculated as:
C = 2πr
C = 2π × 15 cm
C ≈ 94.25 cm
Since the stone completes 2 oscillations in one second, the total distance covered by the stone in one second is given by:
D = 2 × C
D = 2 × 94.25 cm
D ≈ 188.5 cm
The linear speed of the stone is the distance covered per unit time, which is given by:
V = D / T
V = 188.5 cm / 10 s
V ≈ 18.85 cm/s
The angular speed, ω, is defined as the angle covered per unit time, which is given by:
ω = 2π / T
ω = 2π / 10 s
ω = π/5 rad/s
Therefore, the angular speed of the stone is π/5 rad/s and the linear speed is approximately 18.85 cm/s.
To calculate the angular and linear speeds of the stone, we need to use the formulas that relate these quantities to the number of oscillations and the radius of the circular path.
1. Angular Speed (ω):
The angular speed is given by the formula:
ω = (2π * n) / t
where,
ω = angular speed (in radians per second)
n = number of oscillations
t = time taken for the given number of oscillations
From the given information, we know that the stone makes 20 oscillations in 10 seconds. Substituting these values into the formula, we get:
ω = (2π * 20) / 10 = 4π rad/s
2. Linear Speed (v):
The linear speed is given by the formula:
v = r * ω
where,
v = linear speed (in cm/s)
r = radius of the circular path (in cm)
ω = angular speed (in radians per second)
From the given information, we know that the radius of the circular path is 15 cm, and we have already calculated the angular speed as 4π rad/s. Substituting these values into the formula, we get:
v = 15 cm * 4π rad/s = 60π cm/s
So, the angular speed of the stone is 4π rad/s, and the linear speed is 60π cm/s.