Let's break this problem into parts and solve each question step-by-step.

1. Angular displacement in radians and degrees (78 rpm for 5 minutes):

• Revolutions per minute (rpm): $78 \, \text{rpm}$
• Time: $5 \, \text{minutes} = 5 \times 60 = 300 \, \text{seconds}$
• Number of revolutions: $\text{Revolutions} = 78 \, \text{rev/min} \times 5 \, \text{min} = 390 \, \text{revolutions}$
• Angular displacement in radians: Since one revolution equals $2\pi \, \text{radians}$: $\theta = 390 \, \text{rev} \times 2\pi \, \text{radians/rev} = 780\pi \, \text{radians}$ $\theta \approx 2450.44 \, \text{radians}$
• Angular displacement in degrees: Since one revolution equals $360^\circ$: $\theta = 390 \, \text{rev} \times 360^\circ/\text{rev} = 140400^\circ$

Answer:

• Angular displacement: $780\pi \, \text{radians}$ or approximately $2450.44 \, \text{radians}$, and $140400^\circ$.

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2. Linear speeds of points 3 cm and 12 cm from the center:

The linear speed ($v$) is related to angular speed ($\omega$) and radius ($r$) by the formula: $v = r \omega$

• Angular speed ($\omega$): Convert $78 \, \text{rpm}$ to radians per second: $\omega = 78 \, \text{rev/min} \times \frac{2\pi \, \text{radians/rev}}{60 \, \text{seconds/min}} = \frac{156\pi}{60} \approx 8.168 \, \text{radians/second}$

• Linear speed for $r = 3 \, \text{cm}$: $v = 3 \, \text{cm} \times 8.168 \, \text{radians/second} \approx 24.504 \, \text{cm/second}$

• Linear speed for $r = 12 \, \text{cm}$: $v = 12 \, \text{cm} \times 8.168 \, \text{radians/second} \approx 98.016 \, \text{cm/second}$

Answer:

• Linear speed at $3 \, \text{cm}$: $24.504 \, \text{cm/s}$
• Linear speed at $12 \, \text{cm}$: $98.016 \, \text{cm/s}$

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3. Corresponding angular speeds:

Angular speed ($\omega$) is the same for all points on the record, as it rotates as a rigid body.

Answer:

• Angular speed: $\omega = 8.168 \, \text{radians/second}$

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Summary of Results:

1. Angular displacement: $780\pi \, \text{radians}$ or approximately $2450.44 \, \text{radians}$, $140400^\circ$.
2. Linear speeds:
   • At $3 \, \text{cm}$: $24.504 \, \text{cm/s}$
   • At $12 \, \text{cm}$: $98.016 \, \text{cm/s}$
3. Angular speed: $8.168 \, \text{radians/second}$.

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Follow-up Questions:

1. What would be the angular displacement for 10 minutes of rotation at 33 rpm?
2. How does the linear speed change if the radius is doubled?
3. If the record rotates at 45 rpm instead of 78 rpm, what are the new linear speeds for the same radii?
4. How is the angular speed affected if the rotational speed changes from 78 rpm to 33 rpm?
5. What is the relationship between linear speed and radius for a point on a rotating object?

Tip: Always check units when converting between angular and linear quantities to ensure consistency.