1. A cart with mass 3.5 kg is attached to a spring with a spring constant of 630 N/m as shown in the figure below and undergoes a simple harmonic motion on the frictionless surface. The cart is either atx= +22 cm orx= 22 cm when its speed is 0 m/s. The point O in the figure is the equilibrium position, and there is no other force exerted on the cart besides the spring force.

(a)	What is the period of this simple harmonic motion?
(b)	What is themagnitudeand thedirectionof the spring force on the cart at the moment when the spring is stretched by 8.0 cm?
(c)	What is the elastic potential energy at the moment when the cart is atx= 22 cm?
(d)	What is the maximum speed of the cart?
(e)	How far is the cart from the equilibrium position when the speed of the cart is 2.4 m/s?

2. Ultrasound sonography is a tool that is used to image internal organs. In ultrasound sonography, a piezoelectric transducer produces a high-frequency mechanical vibration of a piezoelectric crystal by applying oscillating electric voltage, and the mechanical vibration of the transducer propagates as an ultrasound wave when it is placed on a patient's body. Some of the ultrasound signal reflects back from some parts in the patient's body, and the returning ultrasound signal is detected in the transducer. The distance to the internal organs is calculated by measuring the time it takes the ultrasound to propagate out and back to the transducer. (See "Ultrasound Sonography" on page 522 of the textbook for further details if you are interested.)

The same ultrasound sonography equipment can be used to measure the velocity of a blood flow by incorporating the measurement of Doppler effect (Doppler ultrasound). In essence, Doppler ultrasound measures the frequency of the returning ultrasound signal. The change in the ultrasound frequency is caused by a moving object, i.e., a flow of blood, and the direction and the speed of the blood flow can be determined from how much the frequency is increased or decreased in the returning ultrasound signal.

With this in mind, answer the flowing questions.

An ultrasound signal with frequency 3.8 MHz (3800000 Hz) propagates in fat within a patient's body and reaches the surface of the liver. Some portion of the ultrasound reflects at the interface between the fat and the liver, and the other portion transmits into the liver. The acoustic impedance of the fat is 1330000 kg/m²s, and that of the liver is 1650000 kg/m²s. The density of the fat is 920 kg/m³.

(a)	What is the speed of the ultrasound in the fat?
(b)	What is the distance between the transducer and the surface of the liver in the patient if it takes 52 s for the ultrasound signal to go out and come back to the transducer? Assume that the ultrasound travels mostly in the fat. (2 pt) [note: the distance the ultrasound travels is twice the distance from the transducer to the liver because it is a round trip.]
(c)	What proportion of the ultrasound energy is reflected at the interface between the fat and the liver?

The same frequency of ultrasound is used to measure the velocity of blood flow in an artery within the patient's body. When the ultrasound reflects off flowing blood, the blood can be seen as a moving ultrasound source that emits the ultrasound with the same original frequency of 3.8 MHz. However, the apparent frequency that the detector in the transducer detects is shifted because of the Doppler effect. You can treat the detector in the transducer as a stationary observer. Assume that the speed of sound in the patient body is 1500 m/s for the following two questions.

(d)

When scanned over some part of the artery, the frequency detected in the transducer is decreased by 420 Hz (the apparent frequency = 3799580 Hz). In this case, the blood is moving away from the transducer because the apparent frequency is decreased. What is the speed of the blood flow in this part of the artery?

(e)

The speed of the blood flow slows down when stenosis is present in the blood vessel, and this shows up as a smaller shift in the detected frequency. When scanned over another part of the artery, the detector receives ultrasound frequency with a shift by 230 Hz (the apparent frequency = 3800230 Hz). In this case, the blood is moving toward the transducer because the apparent frequency is increased. What is the speed of the blood flow in this part of the artery?