Oct-12-2016: Lab 14.5 - Ballistic Pendulum

LAB 14.5 - Ballistic Pendulum

Tony Wu, Isaiah Hernandez

October 12, 2016

Lab Goal: Determine the firing speed of a ball from a spring-loaded gun.

Theory/Introduction: In this lab a spring-loaded "gun" fires a ball into a nylon block, which is supported by four vertical strings.  The ball is absorbed into the block, and the ball and block rise together through some angle, which is measured by the angle indicator shown.  The ball, mass m, undergoes an inelastic collision with the nylon block, mass M.  We use conservation of momentum to write an equation for the speed of the system immediately after the collision.  The ball and block system rises, losing kinetic energy and gaining potential energy.  At the system's maximum height, the kinetic is zero.  We use the conservation of energy to relate the maximum height of the block to the initial speed of the block.

Apparatus:

The "gun" shoots the ball that lands in the nylon block which pushes the stick, signifying the angle

Experimental Procedure: First we measured and recorded the mass of the ball and the block.  Then we leveled the base of the apparatus so that the "gun" would shoot the ball directly into the block.  We used the first notch of the gun to fire the ball into the block three times.  We recorded the angle that each trial made and took an average.  We relate kinetic energy to potential energy to solve for the velocity_final.  Then we use the conservation of momentum to relate find the initial velocity.  The calculations are shown in the following section.

Data and Calculations: 

Mass of Ball: 8g = 0.008 kg
Mass of Block: 80g = 0.080 kg

Trial    Angle
#1   === 17.5°
#2   === 17.5°
#3   === 19.5°
Average Angle = 18.2 °

Calculations for Kinetic Energy => Potential Energy

Calculations for Conservation of Momentum in order to find Initial Velocity

The calculations at the bottom are the verification for the accuracy of our calculated value of initial velocity.  We put carbon paper down on the ground and shot the ball out.  The ball is initially shot at a height of 0.995m.  The time it takes for the ball to land on the ground 0.4506 seconds.  The expected distance the ball should have traveled is 2.195 m.  The average recorded distance the ball traveled is 2.382 m.  The difference can be from measurement inaccuracies, rounding errors, air resistance, friction, and/or the collision not being truly inelastic.

Conclusions: 
We were able to relate the angle which the ball+block system made to the potential energy that is gained from the initial kinetic energy.  Then we were able to determine the initial firing speed of the ball from the spring loaded gun. The initial speed is a fairly good estimated value as seen from the comparison of the expected distance the ball traveled vs. the actual distance the ball traveled.

Oct-3-2016: Lab 9 - Centripetal Force with a Motor

LAB 9 - Centripetal Force with a Motor

Tony Wu, Isaiah Hernandez

October 13, 2016

Lab Goal: To observe a spinning mass in order to determine the relationship between an angle theta and angular speed omega.  Theta is the angle that a string makes with the vertical as the mass rotates around a central axis.

Theory/Introduction: A mass is hanging from a rod that is rotated around a central axis by a motor.  In this lab, we vary the speed at which the motor rotates the rod.  The angle theta is the angle that the string makes with the vertical as the mass rotates. The quicker the rod rotates, the greater the angle theta.  We also observe the height h that the mass reaches as it spins.  The greater the angle theta, the greater the height h.  Three constants that we take note of are height H, the top of the string that is attached to the rotating rod, length R, the length of the rotating rod, and the string length L which the mass is attached to.

The overall idea is that given all the variables, we can determine the a relationship between how a change in the angular speed, omega would affect the angle theta, and vice versa.

Apparatus: 

Experimental Procedure: We start off by setting the motor to an arbitrary speed, slow at first. We track the time it takes for the mass to make 10 revolutions.  We determine the height that the mass is from the ground (h) by placing a stand with a small piece of paper close to the where the mass is rotating. Once the mass clips the paper, we record that as h.  We are not able to measure the angle theta but we can calculate it using theta = cos^-1[(H-h)/L].  The lab is recording the different values of the period T (seconds/revolution) as we increase the speed of the motor.  Omega = 2π/T.  

Data and Calculations: 

Recorded Data for Total Time for 10 rotations, Period T in seconds/rotation and Height h in cm.

Freebody Diagram for the Hanging Mass

Calculations that relate theta to omega

Below are the calculations where we plug in values for theta, R, L, and H. 

The only variable left in the equation is h which we will plug in from our recorded data earlier.

Below is the the omega values that we timed, which we got from the equation Omega = 2π/T.
The calculated values are the values we got from our equation that relates H, h, L, R,  and theta to omega.

Correlation of omega_timed vs. omega_predicted (omega_calculated)

Conclusions: As you can see, the values of omega_timed are all generally close to the values of omega_calculated.  The values are all on average within 5% of each other.  The slower speeds, speeds 1-3, all were closer in value than the higher speeds, speeds 4-6. This is likely due to user error, inaccurate estimates of the initial recorded rotations and/or rounding calculation errors. Nonetheless, the derived equation that relates omega and theta is a good estimate.

Sep-28-2016: LAB 8 - Centripetal Acceleration vs. Angular Frequency

LAB 8 - Centripetal Acceleration vs. Angular Frequency

Tony Wu, Leslie Zhao, Isaiah Hernandez

September 28, 2016

Lab Goal: To determine the relationship between centripetal acceleration and angular speed.

Theory/Introduction: Centripetal acceleration is the acceleration of an object towards the center of a circular path.  Centripetal means "center seeking".  An object traveling in a circular path can accelerate even if it's moving at a constant speed.  Acceleration also means change in direction of the velocity.  Therefore, an object traveling in a circle with constantly changing velocity has its acceleration towards the center at any moment.

Apparatus: Large heavy rotating wooden disk, powered by a motor with variable voltages.  The disk rotates at higher velocities as the voltage is increased.  At the center of the disk is a force gauge with a string attached to a mass.  The force gauge reads the tension in the string which is equal to the centripetal force.  The force varies depending on the size of the mass, the length of the string, and the voltage that the disk rotates.

Experimental Procedure: We attached a 200 g mass to a string of length 19 cm and set the voltage to 6.1 volts.  We let the disk speed up until it was running at a roughly a constant speed.  We took note of how many seconds it took for 10 full rotations of the disk.  We then recorded the force reading.  We repeated this process with different masses, lengths of string, and different voltages.

Data and Calculations: 

Recorded Data with Varying Mass, String Length, and Voltage

These graphs show how a change in mass, radius, or omega affects the centripetal force.

Conclusions: The conclusions came out as it would have intuitively.  The equation for centripetal force is: 
F_centripetal = m*r*ω²

An increase in mass means a greater centripetal force. A decrease in mass means a smaller force.  This makes sense as force is mass times acceleration.  An increase in string length, or radius leads to an increase in force, as is verified in our data above.  An increase in voltage, leading to a larger omega value, also leads to a greater force.