Dana loads luggage into an airplane using a conveyor belt tilted at an angle of 20°.She places a few pieces of luggage on the belt before it starts to move, then she turns the belt on. It takes the belt 0.70 s to reach its top speed of 1.2 m/s. Does the luggage slip? Assume a static friction coefficient of 0.5 between the luggage and belt
I really just would like to know what to find i.e. in what order. I have all the formulas. I have found the accel. of the belt and found what the normal force would be equal to but dont know the mass of the luggage
a = acceleration up slope = 1.2/0.7 = 1.714 m/s^2
so net force up slope, m * a, must be 1.714 m
now how
much a do we get maximum
weight force down slope = m g sin 20
normal force = m g cos 20
so max friction force up slope = 0.5 m g cos 20
0.5 m g cos 20 - m g sin 20 = m a
m g ( .470 - .342) = m a
a = .128 g = 1.25 m/s^2
we only have enough friction to get a = 1.25 but we need 1.714
.......but the mass cancels
Well, well, well, if it isn't Dana and her rebellious luggage! Let's try to solve this puzzling problem step by step.
First things first, we need to find the acceleration of the belt. To do that, we can use the equation:
acceleration = change in velocity / time
Given that the belt takes 0.70 seconds to reach its top speed of 1.2 m/s, we can plug in the numbers and calculate the acceleration.
Now, since we're dealing with static friction here (nasty stuff, I tell you), we need to make sure it's strong enough to prevent the luggage from slipping. Static friction can be calculated using the formula:
friction force = static friction coefficient × normal force
Here comes the tricky part. We need to find the normal force acting on the luggage. The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, it's the force exerted by the belt on the luggage.
Since we don't know the mass of the luggage, we can't calculate the weight directly. However, we do know that the normal force is equal to the weight when there's no slipping. So, we can use that knowledge to find the normal force.
Once we've found the normal force, we can calculate the maximum friction force by multiplying it with the static friction coefficient. If the maximum friction force is greater than or equal to the force required to overcome the weight of the luggage, then the luggage won't slip. Otherwise, that luggage is going for a wild ride!
So, in summary, the order of operations would be:
1. Calculate the acceleration of the belt.
2. Find the normal force acting on the luggage (which is equal to the weight when there's no slipping).
3. Calculate the maximum friction force by multiplying the normal force with the static friction coefficient.
4. Compare the maximum friction force with the force required to overcome the weight of the luggage.
5. Determine whether the luggage slips or not.
Hope this helps you solve the problem and prevent any luggage-related circus acts!
To determine whether the luggage slips or not, you'll need to follow these steps:
1. Find the acceleration of the belt:
Since the belt takes 0.70 seconds to reach its top speed of 1.2 m/s, you can use the formula for constant acceleration:
Acceleration (a) = Change in velocity (Δv) / Time (t)
a = (1.2 m/s - 0 m/s) / 0.70 s
Calculate the value of acceleration (a).
2. Determine the normal force:
The normal force (N) is the force exerted by a surface perpendicular to that surface. In this case, as the luggage is placed on the conveyor belt inclined at 20°, the component of the weight of the luggage perpendicular to the belt is equal to the normal force.
Knowing this, you can use the formula:
Normal force (N) = Mass (m) × Acceleration due to gravity (g) × Cosine of the angle (θ)
The angle in this case is 20°, and the acceleration due to gravity is approximately 9.8 m/s².
Calculate the value of the normal force (N).
3. Calculate the maximum frictional force:
The maximum frictional force (Ff) is equal to the product of the normal force and the static friction coefficient. The static friction coefficient is given as 0.5.
Ff = Static friction coefficient (μ) × Normal force (N)
Calculate the value of the maximum frictional force (Ff).
4. Determine the maximum mass of the luggage:
As the luggage starts to slip when the maximum frictional force is reached, you can use the formula:
Force (F) = Mass (m) × Acceleration (a)
In this case, the force is the maximum frictional force (Ff) calculated in the previous step.
Rearranging the formula, you can find the maximum mass (m) of the luggage.
Once you have completed these steps, you should be able to determine whether the luggage slips or not based on the given information.
To determine if the luggage slips, you need to find the force exerted by gravity on the luggage and the maximum force of static friction between the luggage and the conveyor belt. Here's a step-by-step guide on how to approach the problem:
1. Find the acceleration of the conveyor belt:
- Given: Time taken to reach top speed (0.70 s) and top speed (1.2 m/s)
- Acceleration (a) = (Final velocity - Initial velocity) / Time
- Calculate the acceleration of the conveyor belt.
2. Calculate the gravitational force on the luggage:
- Determine the downward force due to gravity acting on the luggage.
- Given: The mass of the luggage (unknown)
- Use the formula: Force (F) = Mass (m) x Acceleration due to gravity (g)
- Calculate the force exerted by gravity on the luggage.
3. Calculate the maximum force of static friction:
- Given: Static friction coefficient (0.5)
- The maximum force of static friction (Fs) is given by the formula Fs = coefficient of static friction * Normal force.
- Determine the normal force acting on the luggage.
4. Determine if the luggage slips:
- If the force exerted by gravity is greater than the maximum force of static friction, the luggage will slip.
- Compare the force exerted by gravity to the maximum force of static friction.
- If the force of gravity is greater, the luggage slips; otherwise, it does not.
5. Solve for the mass of the luggage:
- Unfortunately, the given information in the problem does not provide the mass of the luggage. You need to have the mass to calculate the force exerted by gravity and determine if the luggage slips. If the mass is not provided, you might need to make an assumption or estimate the range of possible masses.
Remember to use the appropriate units throughout the calculations and always be consistent with the unit conversions.