A horizontal spring has one end fixed and one end attached to a 3 kg mass, which slides without friction. The spring has a stiffness constant of 48 N/m. At time t = 0 the mass is at rest at the origin when a driving force of F = 120 cos 6t (F is in newtons) is applied.

(a) Find the natural oscillation frequency, ω◦, and show that the homogeneous solution (i.e., the solution to the motion when F = 0 ) is xh(t) = A cos 4t + B sin 4t.

(b) Use the method of undetermined coefficients to find the particular solution. Use an initial guess of xp(t) = C cos 6t + D sin 6t, then find values for C and D. [Ans: xp(t) = −2 cos 6t]

(c) Apply the initial conditions to show that the general solution is x(t) = 2 [cos 4t − cos 6t].

(d) Use a trig identity to re-write the general solution in part (c) as x(t) = 4 sin t sin 5t

The coordinates of a bird flying in the xy plane are given by x(t)=?t and y(t)=3.0m??t2, where ?=2.4m/s and ?=1.2m/s2

Part A:

Calculate the velocity vector of the bird as a function of time.

Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3t and the y component is 4t, then you should enter 3t,4t. Express your answer using two significant figures for all coefficients.

Part B:

Calculate the acceleration vector of the bird as a function of time.

Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3t and the y component is 4t, then you should enter 3t,4t. Express your answer using two significant figures for all coefficients.

Part C:

Calculate the magnitude of the bird's velocity at t=2.0 s.

Express your answer using two significant figures.

Part D:

Let the direction be the angle that the vector makes with the +x axis measured counterclockwise. Calculate the direction of the bird's velocity at t=2.0 s.

Express your answer in degrees using two significant figures.

Part E:

Calculate the magnitude of the bird's acceleration at t=2.0 s.

Express your answer using two significant figures.

Part F: Calculate the direction of the bird's acceleration at t=2.0 s

Part G: At t=2.0 s, is the bird speeding up, slowing down or moving at constant speed?

A 1.14 kg hollow ball with a radius of 0.133 m, filled with air, is released from rest at depth of 2.09 m in a pool of water. (depth is to the center of ball) What is the net vertical force acting on the ball?(Neglect all frictional effects. Neglect the ball's motion when it is only partially submerged. Neglect the mass of the air in the ball.) What is the work done by the net vertical force on the ball as the ball moves from the bottom of the pool to the surface? How high above the water does the ball shoot upward?

Three point charges, q1, q2, and q3, lie along the x-axis at x = 0.0 cm, x = 3.0 cm, and x = 5.0 cm, respectively. Calculate the electric force on q2 if q1 = +6.0 ?C, q2 = +1.5 ?C, and q3 = -2.0 ?C.

a. Calculate the linear acceleration (in m/s²) of a car, the 0.340 m radius tires of which have an angular acceleration of 11.0 rad/s². Assume no slippage.

b. How many revolutions do the tires make in 2.50 s if they start from rest?

c. What is their final angular velocity (in rad/s)?

d. What is the final velocity (in m/s) of the car?

A rock is projected upward from the surface of the moon y=0, at t=0 with a velocity of 30m/s(j). The acceleration due to gravity at the surface of the moon is -1.62m/s^2(j). When will the rock have a speed of 8.0 m/s?

Julia and Cate want to go bungee jumping. Julia goes first. She has a mass of m and uses a bungee cord with the length of L and unknown springiness. However, Cate is very hesitant to go and wants to calculate how far she will drop before she is pulled back up. She has a mass 0.8 times Julia's mass. How far will Cate go down?

4. A +15 nC point charge is placed on the x axis at x = 1.5 m, and a -20 nC charge is placed on the y axis at y = -2.0m. What is the magnitude of the electric field at the origin?

a) 33.6 N/C b) 64.8 N/C c) 54.4 N/C d) 91.6 N/C e) 74.9 N/C

An object with a charge of -2.9 μC and a mass of 4.6×10⁻² kg experiences an upward electric force, due to a uniform electric field, equal in magnitude to its weight.

A. Find the magnitude of the electric field.

B. Find the direction of the electric field.

C. If the electric charge on the object is doubled while its mass remains the same, find the magnitude of its acceleration.

D. If the electric charge on the object is doubled while its mass remains the same, find the direction of its acceleration.

what needs to be done to charges in order to construct a battery ? what is electric potential and its unit used ?

Tom and his little sister are enjoying an afternoon at the ice rink. They playfully place their hands together and push against each other. Tom's mass is 72 kg and his little sister's mass is 15 kg.

(a) Which of the following statements is correct?

A)The force experienced by Tom is less than the force experienced by his sister.

B)The force experienced by the sister is less than the force experienced by Tom.     

C)They both experience the same force.

(b) Which of the following statements is correct?

A)Tom's acceleration is less than the sister's acceleration.

B)They both have the same acceleration.    

C) Tom's acceleration is more than the sister's acceleration.

(c) If the sister's acceleration is 3.1 m/s² in magnitude, what is the magnitude of Tom's acceleration?
_______m/s²

In the interval between the freezing point (ice point) of water and 700.0deg-C, a platinum resistance thermometer is to be used for interpolating temperatures from 0 °C to the melting point of zinc, 692.666 K. The temperature, in Celcius or Centigrade, is given by a formula for the variation of the resistance of the thermometer with temperature: R = R,o(1 + A Tc + B Tc^2). Ro, A, and B are constants determined by measurements at the ice point, the steam point, and the melting point of zinc. If R equals 12.88350 ohms at the ice point, 88.31639 ohms at the steam point, and 304.28989 ohms at zinc's melting point, find Ro. What is A? What is B?

Modern Physics

A trip is planned to to visit the Alpha Centauri star system 4.3 light years away. Provisions are put in place to allow a trip of 16 years' total duration. How fast must the spacecraft travel if the provisions are to last? Neglect the period of acceleration, turnaround, and visiting times. Express your answer as a multiple of c.

I did the following:

I cut an irregular sheet out of a piece of paper.
I tied the string with a washer at one end to the push pin.
I punched holes in three spot at different "corners" of the paper.
I hung the paper with the push pin from the first hole and let the washer hang down.
I traced the plumb line with a pencil.
I repeated steps 4 and 5 for the other holes.

a) If you cut an irregular shape out of paper and hang it from a thumbtakc it balances around that point it is hung from. If you draw a line straight down the paper from the thumbtack, how is the mass distributed on either side of the line you draw when it is hanging like this?

b) What does the point where the three lines intersect represent? Explain why this method works.

c) Is the third line necessary to find the center of mass? Why or why not?

Two rods, one made of brass and the other made of copper, are joined end to end. The length of the brass section is 0.220m and the length of the copper section is 0.780m . Each segment has cross-sectional area 5.10?10^?3m2 . The free end of the brass segment is in boiling water and the free end of the copper segment is in an ice and water mixture, in both cases under normal atmospheric pressure. The sides of the rods are insulated so there is no heat loss to the surroundings.

1)What is the temperature of the point where the brass and copper segments are joined?

2)What mass of ice is melted in 6.90

min by the heat conducted by the composite rod?