Question 38: Centre of Mass and Rotational Motion

Question

A solid cylinder of mass $$2 \mathrm{~kg}$$ and radius $$0.2 \mathrm{~m}$$ is rotating about its own axis without friction with angular velocity $$5 \mathrm{~rad} \mathrm{s}^{-1}$$. A particle of mass $$1 \mathrm{~kg}$$ moving with a velocity of $$5 \mathrm{~ms}^{-1}$$ strikes the cylinder and sticks to it as shown in figure.

The angular velocity of the system after the particle sticks to it will be

Accepted Answer

The correct answer is
Option C: Angular momentum
. Here's why:
Let's break down the units and their relationships:
Joule (J):
The unit of energy.  Energy is the capacity to do work.
Second (s):
The unit of time.
Angular Momentum (L):
A measure of an object's rotational inertia. It's calculated as the product of the object's moment of inertia (I) and its angular velocity (ÃÂ):
$$L = IÃÂ$$
Now, let's look at the units involved in angular momentum:
Moment of Inertia (I):
Measured in kilogram-meter squared (kgÃÂ·mÃÂ²)
Angular Velocity (ÃÂ):
Measured in radians per second (rad/s)
Combining these, the units of angular momentum are:
$$L = IÃÂ = (kgÃÂ·mÃÂ²) (rad/s) = kgÃÂ·mÃÂ²/s$$
Since radians are dimensionless, we can simplify this to
kgÃÂ·mÃÂ²/s
.  Notice that this is equivalent to
Joule-second (JÃÂ·s)
.
Therefore, Joule-second (JÃÂ·s) is the unit of
angular momentum
.
Let's address why the other options are incorrect:
Option A: Energy
- Energy is measured in Joules (J), not Joule-seconds (JÃÂ·s).
Option B: Power
- Power is the rate at which energy is transferred, measured in Watts (W), which is equivalent to Joules per second (J/s).
Option D: Linear momentum
- Linear momentum is a measure of an object's mass in motion. It's calculated as the product of the object's mass (m) and its velocity (v):
$$p = mv$$
Linear momentum is measured in kilogram-meters per second (kgÃÂ·m/s).

Answer

$$
15.0 \mathrm{~rad~s}^{-1}
$$

Answer

$$
12.0 \mathrm{~rad~s}^{-1}
$$

Answer

$$
10.0 \mathrm{~rad~s}^{-1}
$$

Answer

$$
30.0 \mathrm{~rad~s}^{-1}
$$

Centre of Mass and Rotational Motion

Choose the correct answer: