The system shown below consists of a block of mass M = 4.4 kg resting on a frictionless horizontal ledge. This block is attached to a string that passes over a pulley, and the other end of the string is attached to a hanging block of mass m = 1.9 kg. The pulley is a uniform disk of radius 8.3 cm and mass 0.62 kg. If the (frictionless) ledge were adjustable in angle, at what angle would it have to be tilted upward so that once the system is set into motion the blocks will continue to move at constant speed?
To find the angle at which the blocks will continue to move at constant speed, we need to analyze the forces acting on the system.
1. Find the acceleration of the system:
Since the blocks are moving at a constant speed, the net force on the system must be zero. We can use Newton's second law to find the acceleration:
Net force = Mass of system * Acceleration
The net force acting on the system is the tension in the string, which is equal to the force due to gravity on the hanging block:
Tension = m * g
Where m is the mass of the hanging block and g is the acceleration due to gravity.
The mass of the system is the sum of the masses of the two blocks and the pulley:
Total mass = M + m + mass of pulley
Now we can find the acceleration:
Acceleration = Tension / Total mass
2. Determine the angle at which the blocks will continue to move at constant speed:
The angle at which the blocks will move at a constant speed is the angle at which the force component along the inclined plane is equal to the force due to friction.
The force component along the inclined plane is given by:
Force component = m * g * sin(angle)
The force due to friction can be calculated using the coefficient of friction (µ) and the normal force:
Force due to friction = µ * Normal force
The normal force is equal to the weight of the block on the inclined plane:
Normal force = M * g * cos(angle)
Set the force component equal to the force due to friction and solve for the angle:
m * g * sin(angle) = µ * M * g * cos(angle)
Simplify and solve for the angle:
tan(angle) = µ * M / m
angle = arctan(µ * M / m)
3. Find the coefficient of friction:
To find the coefficient of friction (µ), we need to consider the friction force acting between the block and the inclined plane. Since the ledge is frictionless, there is no friction force.
In this case, the blocks will continue to move at constant speed regardless of the angle of the inclined plane.