Pre-Laboratory Work (estimated time 45 mins) Part I: Problem 1. Consider a cart moving along an incline of angle  with respect to the horizontal plane. a. Suppose you give an initial push to the cart that starts at the bottom of the incline. Your push is in the upward direction along the track. Assume that the incline and the cart are frictionless. Describe the motion of the cart right after the push. b. Is the velocity of the cart constant? Is the acceleration constant? c. The diagram below shows the cart (represented by a box) at initial time (t0) and final time (t3) (the cart is moving from the bottom to the top of the incline). Draw the position of the cart at times t1 and t2, such that the time intervals (t1-t0), (t1-t2) and (t3-t2) are all equal. Also, indicate the velocity and acceleration vectors at each of the four time instants (t0, t1, t2, and t3) in the figure below. d. Suppose that the distance between the initial and the final positions drawn on the previous figure is 0.8 𝑚, the initial velocity of the cart is 0.5 𝑚/𝑠, and (𝑡3 − 𝑡0 ) = 1.3 𝑠. Find the value of 𝑎. Show your work explicitly. e. Find the velocity of the cart when it reaches the final position (at 𝑡3 ). f. Does the acceleration depend on the angle of inclination of the track? If so, how? Date Modified: 04/18/19 Project I Lab 1 – Kinematics Page 1 g. What do you think are the initial experimental conditions that affect the motion of the cart? List them below. Part II: Discretization of the kinematic equations: a. Watch the video “Project 1 video 1” that you can find in E-learning. b. Write down the equation for the numerical evaluation of the derivative of the velocity with respect to time as explained in the video. End of Pre-Laboratory Work Date Modified: 04/18/19 Project I Lab 1 – Kinematics Page 2 Pre-Laboratory Work (estimated time 45 mins) Part I: Problem 1. Consider a cart moving along an incline of angle  with respect to the horizontal plane. a. Suppose you give an initial push to the cart that starts at the bottom of the incline. Your push is in the upward direction along the track. Assume that the incline and the cart are frictionless. Describe the motion of the cart right after the push. The cart will start moving up along the incline. The speed of the cart will go on increasing with time. b. Is the velocity of the cart constant? Is the acceleration constant? The velocity of the cart is not constant (in fact the magnitude of velocity is increasing). The acceleration of the cart is constant. c. The diagram below shows the cart (represented by a box) at initial time (t0) and final time (t3) (the cart is moving from the bottom to the top of the incline). Draw the position of the cart at times t1 and t2, such that the time intervals (t1-t0), (t1-t2) and (t3-t2) are all equal. Also, indicate the velocity and acceleration vectors at each of the four time instants (t0, t1, t2, and t3) in the figure below. 𝑣⃗ 𝑣⃗ t2 𝑣⃗ t1 𝑣⃗ 𝑎⃗ 𝑎⃗ 𝑎⃗ 𝑎⃗ d. Suppose that the distance between the initial and the final positions drawn on the previous figure is 0.8 𝑚, the initial velocity of the cart is 0.5 𝑚/𝑠, and (𝑡3 − 𝑡0 ) = 1.3 𝑠. Find the value of 𝑎. Show your work explicitly. 1 Using second kinematical equation 𝑠 = 𝑢𝑡 + 2 𝑎𝑡 2 we get, (Note that here the time for the motion t is given by t3 – t0 = 1.3 s) 1 2 0.8 = (0.5)(1.3) + 𝑎(1.3)2 ➔ Date Modified: 04/18/19 1 2 0.8 − 0.65 = 𝑎(1.69) ➔ Project I Lab 1 – Kinematics 𝑎 = 0.1775 𝑚/𝑠 2 Page 1 e. Find the velocity of the cart when it reaches the final position (at 𝑡3 ). Using first kinematical equation 𝑣 = 𝑢 + 𝑎𝑡 we get, 𝑣 = (0.5) + (0.1775)(1.3) = 0.73 𝑚/𝑠 f. Does the acceleration depend on the angle of inclination of the track? If so, how? Yes. The acceleration of the cart depends on the angle of inclination of the track. If we increase the inclination, the acceleration of the block will decrease. This is because, as we increase the angle of inclination, the component of weight acting downward along the incline increases. This component opposes the externally applied push which is trying to push the block up the incline. Hence the acceleration of the block decreases as the net force on the block decreases. g. What do you think are the initial experimental conditions that affect the motion of the cart? List them below. 1. The angle of inclination of the plane with horizontal. 2. Presence and magnitude of frictional force – basically the smoothness of the surface of cart and incline. 3. The initial velocity of the cart. 4. The magnitude of the pushing force applied on the cart. Part II: Discretization of the kinematic equations: a. Watch the video “Project 1 video 1” that you can find in E-learning. b. Write down the equation for the numerical evaluation of the derivative of the velocity with respect to time as explained in the video. The yellow colored force represents the external pushing force acting on the block. F The net force along the incline can be calculated as: 𝐹𝑛𝑒𝑡 = 𝐹 − 𝑚𝑔 sin 𝜃 …… (1) (Note that the direction of F and mg sinθ are opposite to each other and hence they will be subtracted mg sin θ θ mg mg cos θ to get resultant) Now the derivative of velocity with respect to time Date Modified: 04/18/19 Project I Lab 1 – Kinematics Page 2 which is nothing but acceleration can be calculated as: 𝑎= 𝐹𝑛𝑒𝑡 𝑚 … (We are using Newton’s Second Law F = ma ➔ a = F/m) Hence we get, 𝑎= 𝐹−𝑚𝑔 sin 𝜃 𝑚 = 𝐹 𝑚 − 𝑔 sin 𝜃 … (Putting the value of Fnet from equation 1) End of Pre-Laboratory Work Date Modified: 04/18/19 Project I Lab 1 – Kinematics Page 3

1. A solid plastic sphere with a radius of 10 cm bears a uniform surface charge density of 360 µC/m2. What is the electric field intensity 1 cm into the sphere from the surface? Assume the permittivity is 9 pC2/N·m2.