Question

A uniform disk with mass 35.2 kg and radius 0.200 m is pivoted at its center about a horizontal, frictionless axle that is stationary. The disk is initially at rest, and then a constant force 34.5 N is applied tangent to the rim of the disk.
a) What is the magnitude v of the tangential velocity of a point on the rim of the disk after the disk has turned through .200 revolution?
b) What is the magnitude a of the resultant acceleration of a point on the rim of the disk after the disk has turned through .200 evolution?

Answer 1

Answer:

a) v = 1.01 m/s

b) a = 5.6 m/s²

Explanation:

a)

• If the disk is initially at rest, and it is applied a constant force tangential to the rim, we can apply the following expression (that resembles Newton's 2nd law, applying to rigid bodies instead of point masses) as follows:

       $\tau = I * \alpha (1)$

• Where τ is the external torque applied to the body, I is the rotational inertia of the body regarding the axis of rotation, and α is the angular acceleration as a consequence of the torque.
• Since the force is applied tangentially to the rim of the disk, it's perpendicular to the radius, so the torque can be calculated simply as follows:
• τ = F*r (2)
• For a solid uniform disk, the rotational inertia regarding an axle passing through its center  is just I = m*r²/2 (3).
• Replacing (2) and (3) in (1), we can solve for α, as follows:

       $\alpha = \frac{2*F}{m*r} = \frac{2*34.5N}{35.2kg*0.2m} = 9.8 rad/s2 (4)$

• Since the angular acceleration is constant, we can use the following kinematic equation:

        $\omega_{f}^{2} - \omega_{o}^{2} = 2*\Delta \theta * \alpha (5)$

• Prior to solve it, we need to convert the angle rotated from revs to radians, as follows:

       $0.2 rev*\frac{2*\pi rad}{1 rev} = 1.3 rad (6)$

• Replacing (6) in (5), taking into account that ω₀ = 0 (due to the disk starts from rest), we can solve for ωf, as follows:

       $\omega_{f} = \sqrt{2*\alpha *\Delta\theta} = \sqrt{2*1.3rad*9.8rad/s2} = 5.1 rad/sec (7)$

• Now, we know that there exists a fixed relationship the tangential speed and the angular speed, as follows:

        $v = \omega * r (8)$

• where r is the radius of the circular movement. If we want to know the tangential speed of a point located on the rim of  the disk, r becomes the radius of the disk, 0.200 m.
• Replacing this value and (7) in (8), we get:

       $v= 5.1 rad/sec* 0.2 m = 1.01 m/s (9)$

b)    

• There exists a fixed relationship between the tangential and the angular acceleration in a circular movement, as follows:

       $a_{t} = \alpha * r (9)$

• where r is the radius of the circular movement. In this case the point is located on the rim of the disk, so r becomes the radius of the disk.
• Replacing this value and (4), in (9), we get:

       $a_{t} = 9.8 rad/s2 * 0.200 m = 1.96 m/s2 (10)$

• Now, the resultant acceleration of a point of the rim, in magnitude, is the vector sum of the tangential acceleration and the radial acceleration.
• The radial acceleration is just the centripetal acceleration, that can be expressed as follows:

       $a_{c} = \omega^{2} * r (11)$

• Since we are asked to get the acceleration after the disk has rotated 0.2 rev, and we have just got the value of the angular speed after rotating this same angle, we can replace (7) in (11).
• Since the point is located on the rim of the disk, r becomes simply the radius of the disk,, 0.200 m.
• Replacing this value and (7) in (11) we get:

       $a_{c} = \omega^{2} * r = (5.1 rad/sec)^{2} * 0.200 m = 5.2 m/s2 (12)$

• The magnitude of the resultant acceleration will be simply the vector sum of the tangential and the radial acceleration.
• Since both are perpendicular each other, we can find the resultant acceleration applying the Pythagorean Theorem to both perpendicular components, as follows:

       $a = \sqrt{a_{t} ^{2} + a_{c} ^{2} } = \sqrt{(1.96m/s2)^{2} +(5.2m/s2)^{2} } = 5.6 m/s2 (13)$