Force on roll of paper - length of paper that unrolls

In summary, the roll of paper has a radius of 7.6 cm and a moment of inertia of 2.9 × 10^-3 kg · m2. A force of 3.2 N is applied for 1.3 s, causing the paper to unroll. A constant friction torque of 0.11 m · N acts on the roll, bringing it to a stop. The paper's thickness is negligible. Using the formulas for torque and angular acceleration, we can calculate the length of paper unrolled during the force application (1.3 s) and after the force ends until the roll stops moving.
  • #1
physicsss
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The radius of the roll of paper is 7.6 cm and its moment of inertia is I = 2.9 10^-3 kg · m2. A force of 3.2 N is exerted on the end of the roll for 1.3 s, but the paper does not tear so it begins to unroll. A constant friction torque of 0.11 m · N is exerted on the roll which gradually brings it to a stop. Assume that the paper's thickness is negligible.


(a) Calculate the length of paper that unrolls during the time that the force is applied (1.3 s).

(b) Calculate the length of paper that unrolls from the time the force ends to the time when the roll has stopped moving.
 
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  • #2
Have you tried it?

Consider these formulas:
[tex]\sum{\tau}=I\alpha[/tex]
[tex]\tau=Fl[/tex]
where [itex]\alpha[/itex] is angular acceleration and [itex]l[/itex] is length of the lever arm.
 
  • #3


a) To calculate the length of paper that unrolls during the 1.3 seconds of force application, we can use the equation for rotational motion: τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. We can rearrange this equation to solve for α: α = τ/I. The force of 3.2 N can be converted to torque by multiplying it by the radius of the roll (0.076 m). Plugging in the values, we get α = (3.2 N)(0.076 m)/(2.9 x 10^-3 kg · m^2) = 0.084 rad/s^2.

Next, we can use the equation for angular displacement to calculate the length of paper that unrolls: θ = ω0t + 1/2αt^2, where ω0 is the initial angular velocity (which we can assume to be 0 since the roll is not initially moving) and t is the time. Plugging in the values, we get θ = 0 + 1/2(0.084 rad/s^2)(1.3 s)^2 = 0.072 rad.

Since 1 revolution = 2π radians, we can convert this to the length of paper unrolled by multiplying by the radius of the roll: (0.072 rad)(0.076 m) = 0.00547 m or 5.47 cm. Therefore, the length of paper that unrolls during the 1.3 seconds of force application is approximately 5.47 cm.

b) To calculate the length of paper that unrolls after the force ends, we can use the equation τ = Iα again. However, this time, we know that the force of friction (0.11 m · N) is acting on the roll, so we can subtract this from the torque: τ = Iα - Ff, where Ff is the force of friction. We can rearrange this equation to solve for α: α = (τ + Ff)/I.

The torque can be calculated using the equation for rotational motion: τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. We can plug in the values and solve for α: α = (0.11 m · N)/(2.
 

1. What is the force on the roll of paper?

The force on the roll of paper is the weight of the paper itself, plus any additional force applied by external factors such as friction or tension.

2. How does the length of paper that unrolls affect the force?

The longer the length of paper that unrolls, the greater the force required to unroll it. This is because there is more paper to move and more resistance from friction.

3. Does the force on the roll of paper change as it unrolls?

Yes, the force on the roll of paper changes as it unrolls. At the beginning, when only a small portion of the paper has unrolled, the force required is relatively low. As more paper unrolls, the force increases due to the weight of the remaining paper and the added friction.

4. How can the force on the roll of paper be calculated?

The force on the roll of paper can be calculated by multiplying the weight of the paper by the acceleration due to gravity (9.8 meters per second squared). This will give you the force in Newtons (N).

5. Are there any other factors that can affect the force on the roll of paper?

Yes, there are several other factors that can affect the force on the roll of paper, such as the type and thickness of the paper, the diameter of the roll, and the surface it is unrolling on. Additionally, any external forces like air resistance or tension can also impact the force required to unroll the paper.

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