A pair of masses are attached by a string that runs over a pulley. The larger mass is on a frictionless incline that makes an angle of 18 degree with the ground. the smaller mass is hanging in the air off the pulley. the larger mass is given an initial velocity of 1.7 m/s up the incline which pulls the smaller mass up by .33 meters after .72 seconds. (the smaller mass is .112 kilograms). what is the mass of the larger masses?
Have you the diagram? Where is the pulley?
Is it at the bottom or at the top of incline?
here is a diagram, with M1 being the larger mass and M2 being the smaller mass.
delete all x's
wxww.csupomona.exdu/~skboddeker/131/notes/pulleyblock.gif
To find the mass of the larger mass, we can start by analyzing the forces acting on the system.
Let's denote:
m1 = mass of the smaller mass
m2 = mass of the larger mass
θ = angle of the incline = 18 degrees
g = acceleration due to gravity = 9.8 m/s^2
μ = coefficient of friction (which is given as zero in this case)
From the information given, we know that the smaller mass is hanging freely and being pulled up by the larger mass on the incline. Therefore, we can set up the following equations:
Equation 1: Tension in the string = m1 * g (since the smaller mass is hanging freely)
Equation 2: m1 * g = m2 * g * sin(θ) (since the weight of m1 is balanced by the component of m2's weight pulling it up the incline)
Equation 3: Fnet = m2 * a (where a is the acceleration of m2)
Now, let's calculate the acceleration of m2 using the information given. We can use the kinematic equation:
Δy = v_initial * t + 0.5 * a * t^2
Substituting the known values:
Δy = 0.33 m
v_initial = 1.7 m/s
t = 0.72 s
0.33 = 1.7 * 0.72 + 0.5 * a * (0.72)^2
0.33 = 1.224 + 0.2592a
0.2592a = 0.33 - 1.224
0.2592a = -0.894
a = -0.894 / 0.2592
a ≈ -3.451 m/s^2 (acceleration of m2)
From Equation 3, we have:
Fnet = m2 * a
Tension in the string - m1 * g = m2 * a
m1 * g = m2 * a
m1 * g = m2 * (-3.451)
m1 * g = -3.451 * m2
Now, substituting the value of m1 (0.112 kg) and solving for m2:
0.112 * 9.8 = -3.451 * m2
1.0976 = -3.451 * m2
m2 = 1.0976 / -3.451
m2 ≈ -0.318 kg
Since mass cannot be negative, there seems to be an error in the given information or calculation. Please review the provided values and calculations to determine the correct mass of the larger mass.
To find the mass of the larger mass, we can use the principles of Newton's laws of motion and equations of motion.
Firstly, let's establish the given information:
- Angle of the incline, θ = 18 degrees
- Initial velocity of the larger mass, v₀ = 1.7 m/s
- Distance the smaller mass is lifted, d = 0.33 meters
- Time taken for the smaller mass to be lifted, t = 0.72 seconds
- Mass of the smaller mass, m = 0.112 kilograms
Now let's proceed step-by-step to find the mass of the larger mass:
1. Determine the acceleration of the system:
The acceleration of the system can be found using the equation of motion:
d = v₀t + (1/2)at²
Substituting the given values, we get:
0.33 m = (1.7 m/s)(0.72 s) + (1/2)a(0.72 s)²
Solving this equation will give us the acceleration (a).
2. Calculate the force causing the acceleration:
The force causing the acceleration is given by:
F = ma
Substitute the mass of the smaller mass (m) and the calculated acceleration (a) to find the force (F).
3. Determine the net force acting on the system:
Since the pulley is frictionless and neglecting any other external forces, we can assume that the net force acting on the system is equal to the force (F) calculated in the previous step.
4. Analyze the forces acting on the larger mass:
The larger mass is on an incline, so the weight of the mass (mg) can be resolved into two components:
- Perpendicular to the incline: mgcosθ
- Parallel to the incline: mgsinθ
The net force on the larger mass (equal to F) is the difference between the parallel component and the force of friction.
5. Calculate the mass of the larger mass:
From step 4, equate the force (F) to the net force on the larger mass and solve for the mass of the larger mass (M).
By following these steps, we can determine the mass of the larger mass in the given scenario.