A wheel of 40cm radius rotates on a stationary axle. It isuniformly speeded up from rest to a speed of 900 rpm in a time of20s. Find (a) the constant angular acceleration of the wheel and(b) the tangential acceleration of a point of a point on itsrim.
a.4.7rad/s^2
b.1.9m/s^2
To find the constant angular acceleration of the wheel, we can use the formula:
\(\text{Angular acceleration} (\alpha) = \frac{\text{Final angular velocity} (\omega_f) - \text{Initial angular velocity} (\omega_i)}{\text{Time taken} (t)}\)
Here, the final angular velocity (\(\omega_f\)) is given in RPM, so we need to convert it to radians per second (rad/s).
Given:
Radius of the wheel (r) = 40 cm = 0.4 m
Final angular velocity (\(\omega_f\)) = 900 RPM
Time taken (t) = 20 s
First, let's convert the final angular velocity to rad/s:
\(\omega_f\) (in rad/s) = 900 RPM x \(\frac{2\pi}{60}\) rad/s = \(\frac{900 \times 2\pi}{60}\) rad/s
Now we can calculate the angular acceleration:
\(\alpha = \frac{\text{Final angular velocity} (\omega_f) - \text{Initial angular velocity} (\omega_i)}{\text{Time taken} (t)}\)
\(\alpha = \frac{\frac{900 \times 2\pi}{60} - 0}{20}\) rad/s²
Now let's calculate the tangential acceleration of a point on the rim of the wheel. Tangential acceleration (\(a_t\)) can be calculated using the formula:
\(a_t = \text{Angular acceleration} (\alpha) \times \text{Radius} (r)\)
Given:
Radius of the wheel (r) = 40 cm = 0.4 m
Angular acceleration (\(\alpha\)) = calculated from above
\(a_t = \alpha \times r\)
\(a_t = \frac{\frac{900 \times 2\pi}{60} - 0}{20} \times 0.4\) m/s²
Now you can calculate the values of (a) the constant angular acceleration and (b) the tangential acceleration of a point on the rim of the wheel.
To find the constant angular acceleration of the wheel, we can use the equation:
Angular acceleration (α) = (final angular velocity - initial angular velocity) / time
Given that the final angular velocity is 900 rpm (revolutions per minute) and the initial angular velocity is 0 (since the wheel starts from rest), we need to convert the final angular velocity to radians per second:
Angular velocity (ω) = (900 rpm) * (2π radians / 1 rev) * (1 min / 60 s) = 30π radians/s
Plugging in the values, we have:
α = (30π radians/s - 0) / 20 s = (30π radians/s) / 20 s = 3π/2 radians/s²
So the constant angular acceleration of the wheel is 3π/2 radians/s².
To find the tangential acceleration of a point on its rim, we can use the equation:
Tangential acceleration (at) = radius * angular acceleration
Given that the radius of the wheel is 40 cm, we need to convert it to meters:
radius = 40 cm = 0.4 m
Plugging in the values, we have:
at = (0.4 m) * (3π/2 radians/s²) ≈ 1.88 m/s²
So the tangential acceleration of a point on the wheel's rim is approximately 1.88 m/s².