the road? What is the minimum coefficient of static friction required to keep the car in this turn? 6. A rotating fan completes 1200 revolutions every minute .Consider a point on the tip of a blade,at a radius of 015 m. (a)through what linear distance does the point move in one revolution? (b)What is the linear speed of the point? 7. The four particles in figure below are connected by rigid rods of negligible mass.The origin is at the center of the rectangle. If the system rotates in the xy plane about the z axis with an angular speed of 6.00rad/s calculate (a)the moment of inertia of the system about the zaxis and (b) the rotational kinetic energy of the system.

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the road? What is the minimum coefficient of static friction required to keep the car in this turn? 6. A rotating fan completes 1200 revolutions every minute .Consider a point on the tip of a blade,at a radius of 015 m. (a)through what linear distance does the point move in one revolution? (b)What is the linear speed of the point? 7. The four particles in figure below are connected by rigid rods of negligible mass.The origin is at the center of the rectangle. If the system rotates in the xy plane about the z axis with an angular speed of 6.00rad/s calculate (a)the moment of inertia of the system about the zaxis and (b) the rotational kinetic energy of the system.

Answer 1

(a) The linear distance the point on the tip of a blade moves in one revolution is 0.94 m. (b) The linear speed of the point on the tip of a blade is 18.8 m/s. (c) The moment of inertia of the system about the z-axis is \(4 m r^2\), where \(m\) is the mass of each particle and \(r\) is the distance of each particle from the z-axis. (d) The rotational kinetic energy of the system is \(\frac{1}{2} \times 4 m r^2 \times (6.00)^2\), where \(m\) is the mass of each particle and \(r\) is the distance of each particle from the z-axis.

Answer 2

## Step 1: Linear Distance Calculation To calculate the linear distance a point on the tip of a blade moves in one revolution, we use the formula for the circumference of a circle, which is given by \(2\pi r\), where \(r\) is the radius of the circle. In this case, the radius is the length of the blade, which is 0.15 m. ### \(d = 2\pi r = 2\pi \times 0.15 = 0.94\) m ## Step 2: Linear Speed Calculation The linear speed of the point can be calculated using the formula \(v = d/t\), where \(d\) is the distance traveled and \(t\) is the time taken. In this case, the distance is the circumference of the circle (0.94 m) and the time is the time taken for one revolution. Since the fan completes 1200 revolutions every minute, the time for one revolution is \(1/1200\) minutes or \(1/20\) seconds. ### \(v = d/t = 0.94 / (1/20) = 18.8\) m/s ## Step 3: Moment of Inertia Calculation The moment of inertia of the system about the z-axis can be calculated using the formula \(I = \sum m r^2\), where \(m\) is the mass of each particle and \(r\) is the distance of each particle from the axis of rotation. In this case, since the origin is at the center of the rectangle, the distance of each particle from the z-axis is the same and is equal to the half of the length of the rectangle. Let's denote this distance as \(r\). The total moment of inertia is then the sum of the moments of inertia of all four particles. ### \(I = 4 m r^2\) ## Step 4: Rotational Kinetic Energy Calculation The rotational kinetic energy of the system can be calculated using the formula \(K = \frac{1}{2} I \omega^2\), where \(I\) is the moment of inertia and \(\omega\) is the angular speed. In this case, the angular speed is given as 6.00 rad/s. ### \(K = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 4 m r^2 \times (6.00)^2\)

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