Wheel Of Radius R And Mass M: Transcribed Image Text

Show Transcribed Image Text A Wheel Of Radius R And Mass M Is Rolling

A wheel of radius r and mass M is rolling on a circular surface of radius R (> r). The wheel rolls without slipping, and the problem involves analyzing its motion as it moves along the circular path. The key considerations include the conditions preventing the wheel from detaching from the surface, especially at the top and bottom points of the circular path, and computing various physical quantities such as velocities, accelerations, and energies at these points.

The problem asks for:

1. The minimum angular velocity of the wheel when it reaches the bottom point B if it does not detach from the surface at the top point A, ensuring it keeps rolling without falling.
2. The velocity and acceleration of the wheel’s center of mass e at the bottom point B under the conditions determined in part (a).
3. The angular velocity and angular acceleration of the wheel at the top point A at the instant it reaches there, under the same conditions.
4. The kinetic energies of the wheel at the top and bottom points of the circular surface.
5. Additionally, the problem hints that the condition for the wheel to stay in contact involves the centrifugal force being no less than gravity.
6. Paper For Above instruction
7. The motion of a rolling wheel on a curved surface involves a combination of translational and rotational dynamics. The analysis begins with setting up the conditions for rolling without slipping and ensuring the wheel remains in contact with the surface at both the top and bottom points. The critical aspect is to determine the minimal initial angular velocity that allows the wheel to reach the bottom without losing contact at the top.
8. Fundamentally, when considering the wheel on a circular path, energy conservation principles can be applied to relate velocities at different points, while the normal force conditions provide constraints for the contact forces. The analysis hinges on the equilibrium of forces in the vertical direction and the requirement that the normal force remains positive.
9. Part a: Minimum Angular Velocity at Bottom Point B
10. To avoid detachment at the top point A, the normal force must remain non-negative throughout the motion. At the top point, the normal force N_A must satisfy:

• Condition for no detachment: N_A ≥ 0

When the wheel reaches the top, the centripetal force requirement and the weight define the normal force as:

N_A = M g - (M v_A^2) / r,

where v_A is the velocity of the wheel’s center at point A. To avoid detachment, N_A ≥ 0, which implies:

(M v_A^2) / r ≤ M g.

Using conservation of energy between the starting point and the top point:

(1/2) M v_B^2 + (1/2) I ω_B^2 = M g h,

considering initial angular velocity ω₀ and the fact that for a rolling wheel:

v = ω r,

and the moment of inertia I for a solid wheel is I = (1/2) M r^2. The energy at the bottom is related to the velocity at the bottom B:

KE at B: (1/2) M v_B^2 + (1/2) I ω_B^2.

At the top point A, the velocity v_A is related to v_B via energy conservation and the height difference.

Integrating these relations, the minimum initial angular velocity ω_min can be derived to satisfy the normal force condition at A, resulting in:

ω_min = √( (g/(r / R)) ), adjusted for the specific geometry and inertia.

Part b: Velocity and Acceleration at Bottom Point B

Applying conservation of energy from initial conditions to the bottom point B gives:

(1/2) M v_B^2 + (1/2) I ω_B^2 = initial energy,

which simplifies to compute v_B because v_B = ω_B r. The acceleration of the center of mass e at B combines tangential and centripetal components, with the centripetal acceleration :

a_c = v_B^2 / R,

and any tangential acceleration depending on energy transfer or external torques.

Part c: Angular Velocity and Acceleration at Top Point A

The angular velocity at A, ω_A, is related to v_A via v_A = ω_A r. The angular acceleration ω̇ depends on torque and angular momentum, but in a conservative system without external torques, angular velocity is constant between points. If acceleration is non-zero, it can be determined from the change in angular velocity over the path segment, considering the energy and force balance.

Part d: Kinetic Energy at Top and Bottom Points

The kinetic energy at the bottom (E_Kb) involves both translational and rotational components:

E_Kb = (1/2) M v_B^2 + (1/2) I ω_B^2.

Similarly, at the top (E_KA):

E_KA = (1/2) M v_A^2 + (1/2) I ω_A^2.

Using energy conservation and the relation between v and ω, the energies can be explicitly computed.

Conclusion

Analyzing the motion of a rolling wheel on a curved surface provides insight into dynamic constraints and energy transformations. Ensuring the wheel remains in contact involves a delicate balance of gravitational, normal, and inertial forces. The minimum initial angular velocity required at the start ensures the wheel reaches the bottom without detaching, with subsequent calculations of velocities, accelerations, and energies following from fundamental principles of mechanics and energy conservation.

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