Is the mechanical energy of the system constant? yes/no b) Is the momentum of the system constant? yes/no c) Is the angular momentum of the system constant? yes/no d) In what direction and with what angular speed does the turntable rotate? – magnitude in rad/s -direction… e) How much chemical energy does the woman’s body convert into mechanical energy of the woman–turntable system as the woman sets herself and the turntable into motion in J?
A 58.0-kg woman stands at the western rim of a horizontal turntable having a moment of inertia of 530 kg · m2 and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim counterclockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to the Earth. Consider the woman–turntable system as motion begins.
a) Is the mechanical energy of the system constant? yes/no
b) Is the momentum of the system constant? yes/no
c) Is the angular momentum of the system constant? yes/no
d) In what direction and with what angular speed does the turntable rotate?
- magnitude in rad/s
-direction…
e) How much chemical energy does the woman’s body convert into mechanical energy of the woman–turntable system as the woman sets herself and the turntable into motion in J?
The Correct Answer and Explanation is :
To analyze the situation, we need to apply concepts from mechanics, particularly those related to rotational motion and conservation laws.
Given:
- Mass of woman (m = 58.0 \, \text{kg})
- Moment of inertia of turntable (I = 530 \, \text{kg} \cdot \text{m}^2)
- Radius of the turntable (r = 2.00 \, \text{m})
- Speed of the woman (v = 1.50 \, \text{m/s}) (relative to the Earth)
- Initial state: Turntable is at rest
a) Is the mechanical energy of the system constant?
Answer: No.
Mechanical energy is the sum of kinetic energy (translational and rotational). As the woman walks, she applies a force to the turntable, causing it to rotate. There is likely some energy lost to friction (between the woman’s feet and the turntable). Hence, mechanical energy is not constant because of the work done against friction.
b) Is the momentum of the system constant?
Answer: No.
The linear momentum of the system (woman + turntable) is not constant. As the woman moves, she imparts a horizontal force to the turntable, causing a change in the system’s center-of-mass velocity. The momentum of the system can change unless external forces (such as friction with the ground) are balanced.
c) Is the angular momentum of the system constant?
Answer: Yes.
The system’s angular momentum is conserved because there are no external torques acting on the woman-turntable system (the friction at the axle is assumed negligible). The woman’s motion induces a counterclockwise torque on the turntable, but the total angular momentum of the system remains constant.
d) In what direction and with what angular speed does the turntable rotate?
- Direction: The turntable rotates clockwise (as viewed from above) to conserve angular momentum. Since the woman walks counterclockwise, the turntable must rotate in the opposite direction to maintain total angular momentum of the system.
- Angular speed: The angular speed can be determined using the conservation of angular momentum. Initially, the total angular momentum is zero because both the woman and the turntable are at rest. As the woman starts walking, her linear momentum generates an angular momentum that must be balanced by the turntable’s opposite angular momentum.
Let’s use the equation for angular momentum:
[
L = I \omega
]
where (I) is the moment of inertia of the turntable, and (\omega) is the angular speed. The woman’s angular momentum is given by:
[
L_{\text{woman}} = m v r
]
Using conservation of angular momentum:
[
L_{\text{woman}} = I \omega
]
Substituting values:
[
58.0 \times 1.50 \times 2.00 = 530 \times \omega
]
[
174 = 530 \times \omega
]
Solving for (\omega):
[
\omega = \frac{174}{530} \approx 0.328 \, \text{rad/s}
]
Thus, the turntable rotates at an angular speed of approximately 0.328 rad/s clockwise.
e) How much chemical energy does the woman’s body convert into mechanical energy?
The work done by the woman to set the system into motion can be related to the change in kinetic energy of the system.
The total kinetic energy of the system consists of:
- The woman’s kinetic energy (translational)
- The turntable’s kinetic energy (rotational)
The woman’s kinetic energy:
[
KE_{\text{woman}} = \frac{1}{2} m v^2 = \frac{1}{2} \times 58.0 \times (1.50)^2 = 130.5 \, \text{J}
]
The turntable’s kinetic energy:
[
KE_{\text{turntable}} = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 530 \times (0.328)^2 = 28.4 \, \text{J}
]
Thus, the total mechanical energy:
[
E_{\text{total}} = 130.5 + 28.4 = 158.9 \, \text{J}
]
The energy converted into mechanical energy is the total kinetic energy of the system, which is approximately 158.9 J. This represents the energy the woman’s body has converted from chemical energy (from the food she consumed) into mechanical energy of the woman-turntable system.