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Math  /  Algebra

Question2) A 68 kg barrel of monkeys is dragged across a floor by pulling on a rope attached to the box and inclined 15∘15^{\circ} above the horizontal. a) If the coefficient of static friction is 0.50 , what minimum force is required from the rope to start the barrel moving? b) If μk=0.35\mu_{k}=0.35, what is the magnitude of the initial acceleration of the barrel?

Studdy Solution

STEP 1

1. The barrel has a mass of 68 kg.
2. The rope is inclined at an angle of 15∘15^{\circ} above the horizontal.
3. The coefficient of static friction (μs\mu_s) is 0.50.
4. The coefficient of kinetic friction (μk\mu_k) is 0.35.
5. We are to find the minimum force required to start the barrel moving and the initial acceleration once it starts moving.

STEP 2

1. Calculate the normal force on the barrel.
2. Determine the force of static friction.
3. Calculate the minimum force required to overcome static friction.
4. Calculate the force of kinetic friction.
5. Determine the net force and calculate the initial acceleration.

STEP 3

Calculate the normal force on the barrel.
The weight of the barrel is given by: W=mg=68 kg×9.8 m/s2=666.4 N W = mg = 68 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 666.4 \, \text{N}
The normal force (NN) is affected by the vertical component of the pulling force. However, initially, we assume the force is just enough to overcome static friction, so we calculate the normal force without the pulling force:
N=Wcos⁡(15∘) N = W \cos(15^\circ)

STEP 4

Determine the force of static friction.
The force of static friction (fsf_s) is given by: fs=μs×N=0.50×666.4cos⁡(15∘) f_s = \mu_s \times N = 0.50 \times 666.4 \cos(15^\circ)
Calculate NN and then fsf_s.

STEP 5

Calculate the minimum force required to overcome static friction.
The minimum force (FF) required to start moving the barrel is equal to the horizontal component of the pulling force that equals the force of static friction:
Fcos⁡(15∘)=fs F \cos(15^\circ) = f_s
Solve for FF: F=fscos⁡(15∘) F = \frac{f_s}{\cos(15^\circ)}

STEP 6

Calculate the force of kinetic friction.
Once the barrel starts moving, the force of kinetic friction (fkf_k) is given by: fk=μk×N=0.35×666.4cos⁡(15∘) f_k = \mu_k \times N = 0.35 \times 666.4 \cos(15^\circ)
Calculate fkf_k.

STEP 7

Determine the net force and calculate the initial acceleration.
The net force (FnetF_{\text{net}}) acting on the barrel is the horizontal component of the pulling force minus the force of kinetic friction:
Fnet=Fcos⁡(15∘)−fk F_{\text{net}} = F \cos(15^\circ) - f_k
Using Newton's second law, calculate the initial acceleration (aa):
Fnet=ma F_{\text{net}} = ma a=Fnetm a = \frac{F_{\text{net}}}{m}
Solution: a) The minimum force required from the rope to start the barrel moving is calculated using the above steps. b) The magnitude of the initial acceleration of the barrel is calculated using the above steps.

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