In Fig. 6-21, a slide-loving pig slides down a certain 37° slide in twice the time it would take to slide down a frictionless 37° slide. What is the coefficient of kinetic friction between the pig and the slide?
down a frictionless slide: mg*SinTheta=ma
down a friction slide: mgSinTheta-mu*mg*CosTheta=ma
so if the time for friction is 2x, then its acceleration is half of the original a.
mgSinTheta-mu*mg*CosTheta=1/2 *(sinTheta)mg
solve for mu.
0.377
To solve this problem, we can use the concept of forces and Newton's second law. We will follow these steps:
1. Draw a free-body diagram of the pig on the slide.
2. Identify the forces acting on the pig. In this case, we have the weight of the pig acting vertically downwards (mg), the normal force (N) acting perpendicular to the slide, and the frictional force (F) acting parallel to the slide.
3. Use Newton's second law to write equations for the forces acting in the y-direction and x-direction.
4. In the y-direction, we have the equation: N - mg*cosθ = 0, where θ is the angle of the slide (37°) and mg*cosθ is the vertical component of the weight.
5. In the x-direction, we have the equation: -F - mg*sinθ = ma, where a is the acceleration of the pig down the slide.
6. Since the pig slides down the frictionless slide in half the time it takes to slide down the slide with friction, we can use the relationship between time and acceleration: t = 2*(2a)^0.5, where t is the time to slide down the slide.
7. Rearrange the equation to solve for a: a = (t^2)/8.
8. Substitute this value of "a" into the equation from step 5 and solve for F: -F - mg*sinθ = (m*t^2)/8.
9. Solve for the coefficient of kinetic friction (μ) by rearranging the equation: F = μ*N.
10. Substitute the value of F from step 8 into this equation and solve for μ: μ*N = mg*sinθ - (m*t^2)/8.
11. Substitute the value of N from step 4 into this equation and solve for μ.
12. Calculate the coefficient of kinetic friction (μ) by dividing both sides of the equation by N.
This step-by-step method will help you solve for the coefficient of kinetic friction between the pig and the slide.
To find the coefficient of kinetic friction between the pig and the slide, we need to use the information provided and apply the principles of physics.
First, let's understand the concept of friction. Friction is a force that opposes the motion between two surfaces in contact. The coefficient of kinetic friction, denoted by µk, represents the ratio between the force of kinetic friction (fk) and the normal force (Fn) between the two surfaces.
The given problem provides the following information:
- The slide has an angle of 37°.
- The pig takes twice as long to slide down this slide compared to a frictionless slide with the same angle.
To solve this problem, we can use the concept of Newton's second law and the relationship between the forces acting on the pig while sliding.
When an object is sliding down an inclined surface, the force that opposes its motion is the force of kinetic friction. The force of kinetic friction (fk) can be determined using the equation:
fk = µk * Fn
Here, Fn is the normal force acting on the object, which can be decomposed into two components: Fn⊥ (perpendicular to the incline) and Fn|| (parallel to the incline).
The perpendicular component, Fn⊥, is responsible for balancing out the component of the object's weight acting perpendicular to the incline. Therefore, Fn⊥ can be expressed as:
Fn⊥ = mg * cos(θ)
where m is the mass of the pig and g is the acceleration due to gravity.
The parallel component, Fn||, is responsible for balancing out the component of the object's weight acting parallel to the incline. Therefore, Fn|| can be expressed as:
Fn|| = mg * sin(θ)
Now, let's apply the given information to solve for the coefficient of kinetic friction (µk).
1. The pig takes twice as long to slide down the slide compared to a frictionless slide with the same angle. This implies that the force of kinetic friction is slowing down the pig's speed. Therefore, the time difference is due to the additional force acting on the pig, which is the force of kinetic friction.
2. Since the slide is frictionless, the only force acting on the pig is its weight (mg) along the incline. This force can be decomposed into two components: mg * cos(θ) perpendicular to the incline and mg * sin(θ) parallel to the incline.
3. The time taken to slide down an inclined plane, neglecting friction, is given by:
t_frictionless = √(2 * d / g * sin(θ))
where d is the vertical distance traveled along the incline.
4. The time taken to slide down the incline with kinetic friction can be expressed as:
t_kinetic_friction = √(2 * d / (g * sin(θ) - µk * g * cos(θ)))
5. The given problem states that t_kinetic_friction is twice t_frictionless. Therefore, we can write:
√(2 * d / (g * sin(θ) - µk * g * cos(θ))) = 2 * √(2 * d / g * sin(θ))
6. Simplifying the above equation, we get:
(g * sin(θ) - µk * g * cos(θ)) = 4 * g * sin(θ)
7. Rearranging the equation, we have:
µk = (g * sin(θ) - 4 * g * sin(θ)) / (g * cos(θ))
8. Simplifying further, we get:
µk = (1 - 4 * sin(θ)) / cos(θ)
Now, substitute the value of θ (37°) into the equation to calculate the coefficient of kinetic friction (µk).