A bicycle wheel is accelerating at the constant rate of 1.4 rev/s2
(a) If it starts from rest, what is the rotational velocity after 4.7s?
(b) Through how many revolutions does it turn in this time?
a.) (1.4)(4.7)=6.58 rev/s
b.) half of 6.58 is the average rev/s as it is constant acceleration so
(3.29)(4.7)=15.463 revolutions
(a) Well, I don't know about you, but that bicycle wheel seems pretty eager to get moving! After 4.7 seconds of consistent acceleration, it will have a rotational velocity of 6.58 rev/s.
(b) As for the number of revolutions it turns in this time, let's do a little calculation. Using the formula v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can find the initial velocity. Since the wheel starts from rest, the initial velocity is zero.
Now, we can use the formula s = ut + 1/2at^2, where s is the displacement or distance traveled. As we know that the initial velocity is zero, the formula simplifies to s = 1/2at^2.
Plugging in the given values, we have s = 1/2 * 1.4 rev/s^2 * (4.7s)^2.
After doing some math, we find that the bicycle wheel turns approximately 18.47 revolutions in that time. So, it's quite the spinning sensation!
To solve this problem, we can use the kinematic equation for rotational motion:
ω = ω₀ + αt
Where:
ω is the final angular velocity
ω₀ is the initial angular velocity (which is 0 in this case because it starts from rest)
α is the angular acceleration (1.4 rev/s²)
t is the time (4.7 s)
(a) To find the rotational velocity after 4.7s, we can substitute the values into the equation:
ω = ω₀ + αt
ω = 0 + (1.4 rev/s²) * (4.7 s)
By solving this equation, we can find ω.
(b) To determine the number of revolutions, we can use the equation:
θ = ω₀t + (1/2)αt²
Where:
θ is the angle in radians
ω₀ is the initial angular velocity (0)
α is the angular acceleration (1.4 rev/s²)
t is the time (4.7 s)
By substituting the values and solving this equation, we can find θ, and then convert it to revolutions.
Let's calculate these step-by-step:
(a) Rotational velocity after 4.7s:
ω = 0 + (1.4 rev/s²) * (4.7 s)
ω = 6.58 rev/s
Therefore, the rotational velocity after 4.7s is 6.58 rev/s.
(b) Number of revolutions in 4.7s:
θ = ω₀t + (1/2)αt²
θ = (0) * (4.7 s) + (1/2) * (1.4 rev/s²) * (4.7 s)²
θ = 6.51 rev
Therefore, the bicycle wheel turns approximately 6.51 revolutions in 4.7 seconds.
To solve these problems, we need to use the equations of rotational motion. Two important equations are:
1. ω = ω0 + αt
2. θ = θ0 + ω0t + 1/2 αt²
Where:
- ω is the final angular velocity
- ω0 is the initial angular velocity
- α is the angular acceleration
- t is the time elapsed
- θ is the final angular displacement
- θ0 is the initial angular displacement
(a) To find the rotational velocity after 4.7 seconds, we need to use equation (1). Given that the wheel starts from rest (ω0 = 0) and the angular acceleration is 1.4 rev/s², we can substitute these values into equation (1):
ω = ω0 + αt
ω = 0 + 1.4 rev/s² × 4.7 s
ω = 6.58 rev/s
So, the rotational velocity after 4.7 seconds is 6.58 rev/s.
(b) To find the number of revolutions made by the wheel in 4.7 seconds, we need to use equation (2). Again, since the wheel starts from rest (ω0 = 0), we can substitute the values into equation (2):
θ = θ0 + ω0t + 1/2 αt²
θ = 0 + 0 + 1/2 × 1.4 rev/s² × (4.7 s)²
θ = 0 + 0 + 1/2 × 1.4 rev/s² × 22.09 s²
θ = 1/2 × 1.4 rev/s² × 22.09 s²
θ = 15.41 rev
So, the wheel turns approximately 15.41 revolutions in 4.7 seconds.