A block of mass m1 = 13.5 kg rests on a wedge of angle θ = 27∘ placed over a table.. An inextensible string is attached to Block M1, passes over a frictionless pulley at the top of the wedge, and is then attached to another block of mass M2 = 3 kg. The coefficient of kinetic friction between M1 and the plane is μ = 0.4. The string and wedge are long enough to ensure neither block hits the pulley or the table in this problem, assume that M1 never reaches the table.The system is released at t= 0
(a) Find the magnitude of the acceleration of block 1 when it is released
(b) How many cm down the plane will block 1 have traveled when 0.475 s has elapsed?
Please tell the steps in brief
To solve this problem, we can use Newton's second law of motion and the principles of kinematics. Here are the step-by-step solutions to the given questions:
(a) Find the magnitude of the acceleration of block 1 when it is released:
Step 1: Draw a free-body diagram for both blocks:
- Block M1: The weight acts vertically downwards, the normal force acts perpendicular to the inclined plane, the frictional force acts opposite to the direction of motion, and the tension force acts upwards.
- Block M2: The weight acts vertically downwards, and the tension force acts upwards.
Step 2: Apply Newton's second law of motion to block M1:
ΣF = ma
mg*sinθ - μ*(mg*cosθ) - T = ma
(m1*g*sinθ - m1*g*μ*cosθ)/(m1 + m2) = a
Substitute the given values: m1 = 13.5 kg, θ = 27º, μ = 0.4, and m2 = 3 kg.
Step 3: Calculate the magnitude of the acceleration, a.
(b) How many cm down the plane will block 1 have traveled when 0.475 s has elapsed:
Step 1: Determine the initial velocity of block M1 when it is released.
Use the equation of motion: v = u + at
Since the block is initially at rest, the initial velocity, u = 0 m/s.
Step 2: Calculate the displacement of block M1 after 0.475 s.
Use the equation of motion: s = ut + 0.5*a*t^2
Substitute the values: t = 0.475 s and a (obtained from part a).
Step 3: Convert the displacement from meters to centimeters by multiplying it by 100.
Note: Make sure to evaluate each step accurately and use consistent units throughout the calculations.
To solve this problem, we can use Newton's second law of motion and the principles of inclined planes. Here are the steps in brief to find the answers:
Step 1: Draw a free body diagram for each block:
For block M1:
- The weight of M1 acts vertically downwards.
- The normal force of M1 acts perpendicular to the incline.
- The friction force acts parallel to the incline, opposing the motion.
- Tension in the string acts upwards and is equal to the weight of M2.
For block M2:
- The weight of M2 acts vertically downwards.
- The tension in the string acts upwards.
Step 2: Write the equations of motion for each block using Newton's second law:
For block M1:
- In the direction parallel to the incline, the net force is given by the difference between the weight component down the incline and the friction force. This is equal to M1 times the acceleration of M1.
- In the direction perpendicular to the incline, the normal force cancels out the component of the weight perpendicular to the incline.
For block M2:
- The net force is given by the difference between the tension in the string and the weight of M2. This is equal to M2 times the acceleration of M2.
Step 3: Express the weight and tension forces in terms of the known quantities.
For M1:
- The weight component down the incline is given by m1 * g * sin(θ), where g is the acceleration due to gravity and θ is the angle of the incline.
- The weight component perpendicular to the incline is given by m1 * g * cos(θ).
For M2:
- The weight force is given by m2 * g.
- The tension in the string is equal to the weight of M2, so T = m2 * g.
Step 4: Substitute these expressions into the equations of motion and solve for the desired quantities.
(a) To find the magnitude of the acceleration of block M1 when it is released, substitute the weight and tension expressions into the equation of motion for M1. Then, solve for the acceleration.
(b) To find how many centimeters down the plane block M1 will have traveled, we need to find the displacement (s) in terms of time (t) and acceleration (a). Use the equation of motion for motion along an inclined plane (s = u * t + (1/2) * a * t^2), where u is the initial velocity (which is 0 in this case because it starts from rest). Substitute the known values and solve for s.
Note: Make sure to convert the time to seconds before substituting into the equations.
These steps should help you solve the problem and find the answers.