A block of mass 20kg is at rest on a rough horizontal ground,the coefficient of friction between the block and the ground is 0.3.A horizontal force is applied steadily to the block according to the law,F= t^2-t+4.calculate the time it takes the block to start moving.
Time? odd question
are you telling us that
force=t^2-t+4? That is very odd indeed.
20*9.8*.3=t^2+t+4
work that out, and solve the quadratic.
Well, well, well, looks like we have ourselves a tricky little physics question, don't we? Don't worry, I'm here to clown around and help you out!
To find the time it takes for the block to start moving, we need to determine the point at which the applied force overcomes the frictional force. Let's dive in, shall we?
The frictional force can be calculated using the equation:
frictional force = coefficient of friction * normal force.
The normal force in this case is simply the weight of the block, which is given by:
weight = mass * acceleration due to gravity.
So, plug in the values, and we get:
weight = 20kg * 9.8m/s² = 196N.
frictional force = 0.3 * 196N = 58.8N.
Now, let's take a look at the applied force equation: F = t² - t + 4.
To find the time it takes for the block to start moving, we need to solve for t when the applied force equals or exceeds the frictional force. So, set up the equation:
t² - t + 4 = 58.8.
Now, this is where things get a little hairy, as we need to solve a quadratic equation. But fear not, my friend! I'm not clowning around when I say we've got to use the quadratic formula:
t = ( -b ± √(b² - 4ac) ) / 2a.
Using the values from our equation, we have:
t = ( -(-1) ± √((-1)² - 4(1)(4 - 58.8)) ) / (2 * 1).
Simplifying further, we find:
t = ( 1 ± √(1 + 235.2) ) / 2.
After some calculation, we have two possible values for t:
t₁ ≈ -6.05.
t₂ ≈ 7.05.
Now, hold on a second! Negative time? Sorry, but that's not possible in this situation. So we're left with t = 7.05 seconds.
So, the time it takes for the block to start moving is approximately 7.05 seconds. Hope that tickles your funny bone!
To calculate the time it takes for the block to start moving, we need to determine the maximum force of friction acting on the block.
The formula for the force of friction can be expressed as:
Frictional force (Ff) = coefficient of friction (μ) * normal force (N)
First, let's calculate the normal force acting on the block:
Normal force (N) = mass (m) * gravity (g)
Given:
Mass (m) = 20 kg
Gravity (g) ≈ 9.8 m/s^2
N = 20 kg * 9.8 m/s^2
N = 196 N
Next, we can determine the maximum force of friction:
Maximum force of friction (Ff_max) = μ * N
Given:
Coefficient of friction (μ) = 0.3
Normal force (N) = 196 N
Ff_max = 0.3 * 196 N
Ff_max = 58.8 N
Now, let's analyze the force applied to the block given by the equation F = t^2 - t + 4. To find the time when the block starts moving, we need to determine when the applied force exceeds the maximum force of friction.
F = t^2 - t + 4
At the point of movement, the maximum force of friction (Ff_max) is equal to the applied force (F):
Ff_max = F
58.8 N = t^2 - t + 4
Now we have a quadratic equation. By rearranging it, we get:
t^2 - t + 4 - 58.8 = 0
Simplifying the equation further, we have:
t^2 - t - 54.8 = 0
Now, we can solve this quadratic equation for t. Let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -1, and c = -54.8.
t = (-(-1) ± √((-1)^2 - 4(1)(-54.8))) / (2(1))
Simplifying further:
t = (1 ± √(1 + 219.2)) / 2
t = (1 ± √(220.2)) / 2
To find the positive root, we have:
t = (1 + √220.2) / 2
Calculating the exact value for t:
t ≈ 8.13
Therefore, it takes approximately 8.13 seconds for the block to start moving.
To find the time it takes for the block to start moving, we need to determine the force needed to overcome the static friction acting on the block.
The formula for static friction is given by:
F_static = μ_s * N
where F_static is the static friction force, μ_s is the coefficient of static friction, and N is the normal force acting on the block.
In this case, the normal force is equal to the weight of the block, which is given by:
N = m * g
where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
So, N = 20 kg * 9.8 m/s^2 = 196 N
Now, we can find the maximum static friction force:
F_static = μ_s * N = 0.3 * 196 N = 58.8 N
Since the applied force is given by F = t^2 - t + 4, we can equate the applied force to the static friction force to find the time it takes for the block to start moving:
t^2 - t + 4 = 58.8 N
Simplifying the equation:
t^2 - t - 54.8 = 0
Now, we can solve this quadratic equation to find the value of t.
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -1, and c = -54.8.
Plugging in the values:
t = (-(-1) ± √((-1)^2 - 4(1)(-54.8))) / (2 * 1)
= (1 ± √(1 + 219.2)) / 2
= (1 ± √220.2) / 2
Now, we have two possible solutions for t:
t_1 = (1 + √220.2) / 2
t_2 = (1 - √220.2) / 2
These are the roots of the quadratic equation. However, we need to consider which one is the correct solution based on the context of the problem.
In this case, since we are looking for the time it takes for the block to start moving, the correct solution is the smallest positive value. After evaluating the two solutions:
t_1 ≈ 10.68 (approximately)
t_2 ≈ -9.68 (approximately)
Therefore, the time it takes for the block to start moving is approximately 10.68 seconds.