A wheel with radius of 25cm turning at 120 rpm uniformly increases its frequency to 660 rpm in 9s. Determine:
i) the centripetal acceleration
ii)the total acceleration of the system
To determine the centripetal acceleration and the total acceleration of the system, we need to understand the equations that relate them to the angular velocity and radius of the wheel.
i) The centripetal acceleration (ac) of an object moving in a circular path is given by the equation:
ac = ω^2 * r
Where:
- ω is the angular velocity in radians per second
- r is the radius of the circular path
ii) The total acceleration (at) of an object moving in a circular path is given by the equation:
at = √(ac^2 + tangential acceleration^2)
Where:
- ac is the centripetal acceleration
- tangential acceleration is the acceleration along the tangent of the circular path
Now let's calculate the values.
Step 1: Convert the wheel's initial and final angular velocities from revolutions per minute (rpm) to radians per second (rad/s).
The conversion factor is: 1 rpm = (2π/60) rad/s
Initial angular velocity:
ω1 = (120 rpm) * (2π/60) rad/s
= 4π rad/s
Final angular velocity:
ω2 = (660 rpm) * (2π/60) rad/s
= 22π rad/s
Step 2: Calculate the change in angular velocity (Δω) and the time taken (t).
Change in angular velocity:
Δω = ω2 - ω1
= 22π rad/s - 4π rad/s
= 18π rad/s
Time taken:
t = 9s
Step 3: Calculate the centripetal acceleration.
Centripetal acceleration:
ac = ω1^2 * r
= (4π rad/s)^2 * (0.25m)
= 16π^2 * 0.25 m/s^2
≈ 12.566 m/s^2
Step 4: Calculate the tangential acceleration.
Tangential acceleration:
At the initial angular velocity (ω1), the tangential acceleration is zero since the speed is constant.
Step 5: Calculate the total acceleration.
Total acceleration:
at = √(ac^2 + tangential acceleration^2)
= √((12.566 m/s^2)^2 + 0^2)
≈ 12.566 m/s^2
Therefore:
i) The centripetal acceleration is approximately 12.566 m/s^2.
ii) The total acceleration of the system is approximately 12.566 m/s^2.
Please note that these calculations assume the wheel is moving in a perfect circle and neglect factors like external forces or any potential slip between the wheel and the surface it is rolling on.