Ramp Problem, Friction, Find Theta

In summary, this problem involves two blocks of masses 2m and m connected by a weightless string over a frictionless, massless pulley. The goal is to find the incline angle \theta at which the blocks move at a constant speed. By analyzing the forces acting on each block, we can set up equations and use trigonometric identities to solve for \theta. It is important to consider the cases of upward and downward motion and use a simple physical picture to rationalize the solutions.
  • #1
maryhem
1
0
Challenging Ramp and Pulley Problem

Homework Statement



Two blocks of masses 2m and m are connected by a weightless string over a frictionless, massless pulley, as shown in the figure. The coefficient of kinetic friction between the block and the incline is [itex]\mu[/itex]. The system is in a uniform gravitational field directed downward of strength [itex]g[/itex]. Find the incline angle [itex]\theta[/itex] such that the blocks move at a constant speed. Distinguish between the cases of upward and downward motion. Rationalize your solutions using a simple physical picture.

Homework Equations


[tex]\mathbf{F} = m\mathbf{a} [/tex]

The Attempt at a Solution



So we start by looking at the forces acting on each block. In this case, we will be looking at downward motion. For the [itex]2m[/itex] mass:
[tex] 2mg\sin\theta - \mu mg\cos\theta - T = 0 [/tex]
And for the second block:
[tex] T - mg = 0 \implies T = mg [/tex]
Using the second equation and plugging into the first equation, we find:
[tex] 2\sin\theta - 2\mu\cos\theta = 1 [/tex]
I can't figure out how to solve for [itex]\theta[/itex]. Wolfram's answer is pretty ugly.
 

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  • #2
You need trig identities and a bit of manipulation:

Divide through by ##\cos(\theta)##
##1/\cos(\theta)=\sec(\theta)##

Square both sides and expand the RHS
##\sec^2(\theta)=1+\tan^2(\theta)##

Change variables: ##x=\tan(\theta)##
Look familiar?

Note - you have misplaced a minus sign in the first equation.
gravity and friction both point in the opposite direction to tension.
 

1. What is the ramp problem and why is it important?

The ramp problem is a physics concept that involves calculating the motion of an object sliding or rolling down an inclined surface. It is important because it helps us understand how friction and gravity affect the movement of objects and allows us to make predictions about their behavior.

2. How does friction affect the ramp problem?

Friction is the force that opposes motion between two surfaces in contact. In the case of the ramp problem, friction plays a significant role in slowing down the object's movement as it slides or rolls down the ramp. The amount of friction depends on the roughness of the surfaces and the force pushing the object down the ramp.

3. How can I calculate the angle of the ramp (theta) in the ramp problem?

The angle of the ramp can be calculated using the formula tan(theta) = height/length. This means that the tangent of the angle is equal to the height of the ramp divided by its length. By taking the inverse tangent of this value, you can find the angle (theta) of the ramp.

4. How does the mass of the object affect the ramp problem?

The mass of the object affects the ramp problem by influencing the amount of force needed to move the object down the ramp. A heavier object will require more force to overcome the force of friction and gravity, while a lighter object will require less force. This can also impact the angle of the ramp needed for the object to slide or roll down without coming to a stop.

5. What are some real-life applications of the ramp problem?

The ramp problem has many real-life applications, such as understanding the movement of objects on ramps or slopes, designing roller coasters and other amusement park rides, and calculating the trajectory of projectiles. It also has practical applications in engineering, such as designing wheelchair ramps and loading ramps for trucks.

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