Question

A stationary bicycle wheel of radius $0.9m$is mounted in the vertical plane (figure). The axle is held up by supports that are not shown, and the wheel is free to rotate on the nearly frictionless axle. The wheel has mass all $4.8kg$concentrated in the rim (the spokes have negligible mass). A lump of clay with mass $0.5kg$falls and sticks to the outer edge of the wheel at the location shown. Just before the impact the clay has speed $5m/s$and the wheel is rotating clockwise with angular speed$0.33rad/s.$

(a) Just before the impact, what is the angular momentum (magnitude and direction) of the combined system of wheel plus clay about the center C? (As usual, is to the right, is up, and is out of the screen, towards you) (b) Just after the impact, what is the angular momentum (magnitude and direction) of the combined system of wheel plus clay about the center C? (c) Just after the impact, what is the angular velocity (magnitude and direction) of the wheel? (d) Qualitatively, what happens to the linear momentum of the combined system? Why? (1) The downward linear momentum decreases because the axle exerts into angular momentum. (3) some of the linear momentum is changed into energy. (4) There is no change because linear momentum is always conserved.

Short answer

The angular momentum of the whole system is$-0.323kg·m^2/s$and the direction is out of the page.

The angular velocity of the wheel after impact is $-0.75rad/s$.

Step-by-step solution

Definition of Angular momentum.

The rotating analogue of linear momentum is angular momentum (also known as moment of momentum or rotational momentum). Because it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics.

The product of the moment of inertia and angular velocity is angular momentum, which is an important feature of a rotating object.

Figure shows the linear velocity and angular velocity of the wheel.

The wheel's linear and angular velocity are depicted in the diagram below.

About the linear momentum and angular momentum of the object.

1. Linear momentum of the particle is expressed as,

$L=m(v\times r)$

Here,$m$is mass,$v$is velocity, and$r$is the position vector.

Since the clay has the linear momentum before it touches the wheel and rotates with it, the linear momentum of the clay from the figure is given as,

$L_{clay}=mvR\cos 45°$

Here,$R$is the radius of the wheel. Negative sign indicates the direction.

Substitute $0.5kg$for $m,0.9m$and $5m/s$for $v$.

$L_{clay}=-\left(0.5kg\right)\left(5m/s\right)\left(0.9m\right)\cos 45°$

$=-1.5909kg·m^2/s$

The angular momentum of the wheel is given as,

$L_{wheel}=I\omega$

Here, $I$is moment of inertia and $\omega$is the angular velocity.

Find the angular momentum of the whole system.

Moment of $I$inertia is expresses as,

$I=M{R}^2$

Here, $M$is the mass of the wheel and $R$is the radius.

Substitute $M{R}^2$for $I$in equation $L_{wheel}=-I\omega$.

$L_{wheel}=M{R}^2\omega$

Substitute $4.8kg$for localid="1668593889199" $M,0.9m$for $R$, and $0.33rad/s$for $\omega$.

$L_{wheel}=\left(4.8kg\right){\left(0.9m\right)}^2\left(0.33rad/s\right)$

$=1.2830kg·m^2/s$

Before the impact of the aforementioned estimated values, the initial angular momentum of the entire system is given as,

$L_i=L_{clay}+L_{wheel}$
$=1.5909kg·m^2/s+\left(1.2830kg·m^2/s\right)$

$=-0.323kg·m^2/s$

Therefore, the angular momentum of the whole system is $-0.323kg·m^2/s$and the direction is out of the page.

(b) The angular momentum of the clay just after the impact is given as,

$L_{clay}=m{r}^2\omega$

The final angular momentum of the whole system is given as,

$L_{system}=L_{clay}+L_{wheel}$

Substitute role="math" localid="1668594915315" $m{R}^2\omega$for $L_{wheel}$and $m{R}^2\omega$for $L_{clay}.$

$L_f=m{R}^2\omega +M{R}^2\omega$

$=\left(m+M\right)R^2\omega$

Apply the law of conservation of angular momentum by which the initial angular momentum before the impact and the final angular momentum after the impact is same.

$L_f=L_i$

$\left(m+M\right)R^2\omega =0.323kg·m^2/s$

Therefore, the final angular momentum of the system just after the impact is$-0.323kg·m^2/s$ and the direction is out of the page.

Find the angular velocity of the wheel after impact.

(c) Apply the law of conservation of energy to find out the angular velocity of the wheel after impact

$L_f=L_i$

$\left(m+M\right)R^2\omega =L_i$

Re-arrange the above equation to solve for $\omega$.

$\omega =\frac{L_i}{\left(m+M\right)R^2}$

Substitute $4.8kg$for $M$$0.9m$for $R$, and $-0.323kg·m^2/s$for $L_i$

$\omega =\frac{-0.323kg·m^2/s}{\left(0.5kg+4.8kg\right){\left(0.9m\right)}^2}$

$=-0.75rad/s$

Therefore, the angular velocity of the wheel after impact is$-0.75rad/s$.

(d) The linear momentum of combined system decreases because the axle exerts an upwards force.

Hence, the correct option is (1).