Question

data collection and analysis
trial | $m_b$ (kg) | $m_h$ (kg) | $f_g$ (n) | radius (m) | time (20 rotations) (s) | period (s) | velocity (m/s) | $f_c$ (n)
a1. | 0.0125 | 5 | 5 | 0.5 | 11.47 | 0.57 | 5.51 |
a2. | 0.0125 | 100 | 100 | 0.5 | 11.16 | 0.55 | 5.71 |
a3. | 0.0125 | 200 | 200 | 0.5 | 9.33 | 0.46 | 6.82 |
a4. | 0.0125 | 500 | 500 | 0.5 | 8.15 | 0.40 | 7.85 |
a5. | 0.0125 | 1000 | 1000 | 0.5 | 7.03 | 0.35 | 8.97 |
b1. | 0.0125 | 0.2 | 0.2 | | | | |
b2. | 0.0125 | 0.2 | 0.2 | | | | |
b3. | 0.0125 | 0.2 | 0.2 | | | | |
b4. | 0.0125 | 0.2 | 0.2 | | | | |
b5. | 0.0125 | 0.2 | 0.2 | | | | |
post lab discussion:

1. draw free body diagrams for the hanging mass ($m_h$) and the bob ($m_b$).
2. explain why an object moving in a circular path has a constant speed but is accelerating at the same time.

Explanation:

For the hanging mass (\(m_h\)): It has two forces - gravitational force (\(F_g = m_hg\)) downward and tension force (\(T\)) upward. If it's in equilibrium (constant velocity or at rest), \(T = F_g\). For the bob (\(m_b\)) moving in a circle: It has tension force (\(T\)) at an angle (or horizontal component providing centripetal force) and gravitational force (\(F_g = m_bg\)) downward. The horizontal component of tension provides the centripetal force (\(F_c=\frac{m_bv^2}{r}\)) for circular motion, and the vertical component balances gravity (\(T_y = m_bg\)) if the setup is such that the vertical motion is negligible.

For the second question: Speed is a scalar (magnitude of velocity). In circular motion, if the object covers equal arcs in equal time, speed is constant. But velocity is a vector (has magnitude and direction). The direction of velocity at any point in circular motion is tangent to the circle, so as the object moves, the direction of velocity changes continuously. Acceleration is the rate of change of velocity (either magnitude, direction, or both). Since the direction of velocity changes, there is an acceleration (centripetal acceleration, \(a_c=\frac{v^2}{r}\)) directed towards the center of the circle, even if the speed (magnitude) is constant.

Answer:

1. Free Body Diagrams:

• Hanging Mass (\(m_h\)):
• Downward: Gravitational force (\(F_g = m_h g\)).
• Upward: Tension force (\(T\)) from the string.

(If \(m_h\) is in equilibrium, \(T = m_h g\).)

• Bob (\(m_b\)):
• Downward: Gravitational force (\(F_g = m_b g\)).
• Tension force (\(T\)) from the string, acting at an angle (or horizontally, depending on setup). The horizontal component of \(T\) provides the centripetal force (\(F_c=\frac{m_b v^2}{r}\)) for circular motion; the vertical component balances \(m_b g\) (if vertical motion is negligible).

1. Circular Motion Explanation:

Speed (scalar, \(v=\frac{\text{distance}}{\text{time}}\)) remains constant if the object covers equal arc lengths in equal time. However, velocity (vector, \( \vec{v} \)) has a direction tangent to the circular path. As the object moves, the direction of \( \vec{v} \) changes continuously. Acceleration is the rate of change of velocity (\( \vec{a}=\frac{\Delta \vec{v}}{\Delta t} \)). Since the direction of \( \vec{v} \) changes, there is an acceleration (centripetal acceleration, \( a_c=\frac{v^2}{r} \)) directed toward the center, even with constant speed.