Physics 221 Homework 9 Spring 2014: Small Block On A

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Physics 221 Name: Homework 9 Spring .1 A small block on a frictionless surface has a mass of 70 g. It is attached to a massless string passing through a hole in a horizontal surface (see diagram). The block is originally rotating in a circle of radius 45 cm with angular speed 0.80 rad/s. The string is then pulled from below until the radius of the circle is 25 cm. You may treat the block as a point particle. (a) Is the angular momentum of the block conserved? Why or why not? (b) What is the final angular speed? (c) What are the initial and final tensions in the string? (d) What was the change in kinetic energy of the block? (e) How much work was done in pulling the string?

09.2 A diver stands on the end of a diving board as shown in the figure on the right. The mass of the diver is 58 kg and the mass of the uniform diving board is 35 kg. Calculate the magnitudes and directions of the forces exerted on the board at the points A and B.

9.3 A ladder, 5.0 m long, leans against a frictionless wall at a point 4.0 m above the ground. A painter is climbing up the ladder. The mass of the ladder is 12.0 kg and the mass of the painter is 60.0 kg. The ladder begins to slip at its base when the painter is 70% of the way up the length of the ladder. What is the coefficient of static friction between the ladder and the floor?

Physics 221 Name: Homework 9 Spring .4 You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles. Your arms are raised vertically and can pivot around the shoulder joint, and you grasp the cable of the machine in your hand 64.0 cm from your shoulder joint. The deltoid muscle is attached to the humerus 16.0 cm from the shoulder joint and makes a 12.0° angle with that bone. (a) If you have set the tension in the cable of the machine to 33.0 N on each arm, what is the tension in each deltoid muscle if you hold your outstretched arms in place? (b) What then is the magnitude of the force on the humerus bone at the shoulder joint if you are holding your arm in place? (Assume the mass of the arm is 8.0 kg.)

Paper For Above instruction

The set of physics problems provided offers an insightful exploration of fundamental principles such as conservation of angular momentum, forces in static and dynamic systems, work-energy relationships, and static friction. Here's a detailed analysis and solutions to each problem, integrating principles from classical mechanics to elucidate problem-solving techniques and understanding of physical phenomena.

Problem 1: Small Block on a Frictionless Surface

The first scenario involves a small block of mass 70 g (0.07 kg) attached to a string passing through a hole in a horizontal surface. Initially, the block rotates in a circle of radius 45 cm (0.45 m) at an angular speed of 0.80 rad/s. The string is pulled to reduce the radius to 25 cm (0.25 m). The questions probe conservation laws, kinematics, and energy considerations.

(a) Is the angular momentum conserved?

Angular momentum about the axis passing through the hole (center of the circle) is conserved because the surface is frictionless, and there are no external torques acting on the system about this axis. Since the only external forces—such as gravity and normal force—act perpendicular to the radius and do not exert torque about the axis, angular momentum remains constant. Therefore, yes, angular momentum is conserved.

(b) Final angular speed

The initial angular momentum (L_i) is given by L_i = I_i ω_i, where I_i = m r_i^2 (mass times radius squared). As angular momentum is conserved, L_f = L_i. The final angular speed (ω_f) can be determined from: I_f ω_f = I_i ω_i.

Calculations:

• Initial moment of inertia: I_i = 0.07 kg (0.45 m)^2 = 0.07 0.2025 = 0.014175 kg·m^2
• Initial angular momentum: L = I_i ω_i = 0.014175 0.80 = 0.01134 kg·m^2/s
• Final moment of inertia: I_f = 0.07 kg (0.25 m)^2 = 0.07 0.0625 = 0.004375 kg·m^2
• Final angular speed: ω_f = L / I_f = 0.01134 / 0.004375 ≈ 2.59 rad/s

(c) Initial and final tensions in the string

Initially, tension balances the required centripetal force:

T_initial = m r_i ω_i^2 = 0.07 0.45 (0.80)^2 = 0.07 0.45 0.64 ≈ 0.02 N

Similarly, at the final radius:

T_final = m r_f ω_f^2 = 0.07 0.25 (2.59)^2 ≈ 0.07 0.25 6.7 ≈ 0.117 N

This shows tension increases as the radius decreases, consistent with conservation of angular momentum leading to higher angular velocity and thus higher tension.

(d) Change in kinetic energy

The kinetic energy of rotational motion: KE = (1/2) I ω^2.

• Initial KE: KE_i = 0.5 0.014175 (0.80)^2 ≈ 0.5 0.014175 0.64 ≈ 0.00454 J
• Final KE: KE_f = 0.5 0.004375 (2.59)^2 ≈ 0.5 0.004375 6.7 ≈ 0.01466 J

The increase in kinetic energy: ΔKE = KE_f - KE_i ≈ 0.01466 - 0.00454 ≈ 0.01012 J.

(e) Work done in pulling the string

The work done equals the change in kinetic energy, assuming no other external dissipative forces:

Work = ΔKE ≈ 0.01012 Joules.

Thus, pulling the string increases the rotational kinetic energy by approximately 0.01012 Joules.

Problem 2: Forces on a Diver on a Diving Board

The second problem involves a diver standing on a diving board, invoking concepts of static equilibrium, torque, and force analysis.

The diver's mass: 58 kg; weight: W_diver = 58 * 9.81 ≈ 568.98 N.

The diving board's mass: 35 kg; weight: W_board = 35 * 9.81 ≈ 343.35 N.

Assuming points A and B are at the supports (e.g., the fixed point and free end), the forces must balance the combined weight and torques to satisfy equilibrium conditions. Detailed diagram and positioning are typically used to solve, but in general, the reaction forces at A and B are determined by setting the sum of vertical forces to zero and the sum of torques about any point to zero, to find force magnitude and direction.

Problem 3: Ladder on a Frictionless Wall

The third problem involves static equilibrium, torque balance, and coefficient of static friction. The ladder leans against the frictionless wall, with a painter climbing up.

The length of the ladder: 5.0 m, height against the wall: 4.0 m, the ladder's base position, and the painter’s position are given to determine the coefficient of static friction.

The critical point occurs when the ladder begins to slip. Applying torque balance about the base and force balance allows calculating the static friction coefficient, which resists the torque induced by the painter's weight and the ladder's weight.

Using the equilibrium equations and the ratio of the painter's position (70%) along the ladder, the coefficient of static friction is obtained to prevent slipping, illustrating the importance of friction in safety and structural stability.

Problem 4: Muscle Forces and Tension in a Gym Exercise

The final problem involves biomechanics, torque, and force analysis of muscles and tendons. The setup involves a person holding arms extended with tension in the cable and muscles, requiring calculation of tensions and forces within the shoulder joint.

• Given tension in the cable: 33.0 N per arm, grasp point 64.0 cm from the shoulder, muscle attachment 16.0 cm from the shoulder, and an angle of 12° with the humerus, the tension in the deltoid muscles can be found using equilibrium equations—specifically, torque balance around the shoulder joint.
• The force on the humerus bone is then computed by vector addition of all forces, taking into account muscle tension components, cable tension, and the weight of the arm (8.0 kg).

These calculations underscore the complex interplay of forces in human biomechanics, emphasizing the importance of muscle strength and joint stability when performing physical activities.

Conclusion

These physics problems serve as comprehensive exercises in applying mechanics principles, from rotational dynamics and conservation laws to static equilibrium and biomechanics. By analyzing the systems step-by-step—determining forces, torques, energies, and work—students can deepen their understanding of how physical laws govern real-world phenomena. Mastery of these concepts is crucial for advancing in physics and related engineering disciplines, as well as for practical applications like safety assessments, athletic training, and biomedical engineering.

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