This is the info for the problem: A block of mass M1 = 160.0 kg sits on an inclined plane and is connected to a bucket with a massless string over a massless and frictionless pulley. The coefficient of static friction between the block and the plane is ms = 0.52, and the angle between the plane and the horizontal is q = 47°. Mass M2 can be changed by adding or taking away sand from the bucket.
It has 3 different parts. The first two I was able to solve. Part 3 asks: Suppose the coefficient of kinetic friction between the block and the inclined plane is muk = 0.36. If M2 = 217.1 kg, what is the magnitude of the acceleration of M1?
Can someone please explain how to do this?
Write two equations, which should be the equation of motion of the two separate bodies: block and bucket. There will be an unknown string tension force T is each equation, acting in opposite directions, but the unknown acceleration "a" will be the same for both bodies. Therefore you will have two equations in two unknowns and can solve for a.
To solve this problem, we need to apply Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration (F = m * a).
In this case, we have two masses connected by a string over a pulley, so they are subject to the same acceleration. Let's call the acceleration of both masses 'a'. We will consider the positive direction to be downwards for Mass 1 and upwards for Mass 2.
Now, let's break down the forces acting on Mass 1:
1. Weight (W1): The force due to gravity acting on Mass 1 can be calculated by multiplying its mass (M1 = 160.0 kg) by the acceleration due to gravity (9.8 m/s^2). We get W1 = M1 * g = 160.0 kg * 9.8 m/s^2.
2. Normal force (N1): The component of the weight perpendicular to the inclined plane is counteracted by the normal force exerted by the plane. N1 = W1 * cos(q), where q is the angle between the plane and the horizontal (47°).
3. Frictional force (F1): The frictional force can be either static or kinetic, depending on whether the block is stationary or in motion. In this case, we are given the coefficient of kinetic friction (muk = 0.36), so we are dealing with kinetic friction. The frictional force can be calculated as F1 = muk * N1.
Now, let's consider the forces acting on Mass 2:
1. Weight (W2): The force due to gravity acting on Mass 2 can be calculated by multiplying its mass (M2 = 217.1 kg) by the acceleration due to gravity (9.8 m/s^2). We get W2 = M2 * g = 217.1 kg * 9.8 m/s^2.
Since both masses have the same acceleration, we can write the following equation using Newton's second law:
Net force on Mass 1 = Mass of Mass 1 * Acceleration (F1 - Tension) = M1 * a
Net force on Mass 2 = Mass of Mass 2 * Acceleration (W2 - Tension) = M2 * a
To solve for the acceleration (a), we need to find the tension in the string.
Tension (T) can be found by considering the vertical force balance for Mass 2:
Tension = W2 - W2 * cos(q)
Now we have all the necessary information to solve the problem. Plug in the known values and solve for the acceleration (a) using the equations derived above.