A bicyclist starts from rest and accelerates at a constant rate for 15.0s. After 10.0s the bicycles front wheel is turning at a rate of 2.5 rev/s. The tire wheel has a diameter of 0.8m. A)Through how many revolutions does the wheel turn in the first 10.0s? B)What is the linear speed of the bicycle at 10.0s? C)How far down the linear acceleration of the cyclist at 10.0s?
2•π•n=ε•t,
ε = 2•π•n /t=2•π•2.5 /10= π/2 rad/s²,
2•π•N=ε•t²/2,
N= ε•t²/4•π= π•100/2•4•π=12.5 rev.
v=2•π•n•R=2•π•2.5•0.4 =2•π m/s,
a= ε•R= π•0.4/2=0.2•π m/s²
To solve these questions, we will use the formulas for linear speed and acceleration.
A) To find the number of revolutions the wheel completes in the first 10.0s, we can use the formula:
Number of revolutions = Angular velocity / (2π)
Given that the front wheel is turning at a rate of 2.5 rev/s, we can calculate the number of revolutions:
Number of revolutions = 2.5 rev/s * 10.0s = 25 revolutions
Therefore, the wheel completes 25 revolutions in the first 10.0s.
B) To find the linear speed of the bicycle at 10.0s, we need to convert the angular velocity into linear velocity. The angular velocity is given in revolutions per second, and we need to convert it to meters per second.
The circumference of the wheel can be calculated using the formula:
Circumference = π * diameter
Given that the diameter is 0.8m, we have:
Circumference = π * 0.8m = 2.52m
Now, we can calculate the linear velocity using the formula:
Linear velocity = Angular velocity * Circumference
Substituting the values:
Linear velocity = 2.5 rev/s * 2.52m/rev = 6.3 m/s
Therefore, the linear speed of the bicycle at 10.0s is 6.3 m/s.
C) To find the linear acceleration of the cyclist at 10.0s, we need to use the formula:
Linear acceleration = (Final linear velocity - Initial linear velocity) / Time
Since the bicyclist starts from rest, the initial linear velocity is 0 m/s. We already found the linear velocity at 10.0s to be 6.3 m/s. So, using the formula:
Linear acceleration = (6.3 m/s - 0 m/s) / 10.0 s = 0.63 m/s²
Therefore, the linear acceleration of the cyclist at 10.0s is 0.63 m/s².