A body of mass 5kg is initially at rest at the bottom of a rough inclined plane of length 6.3m. The plane is inclined at 300 to the horizontal and coefficient of friction between the body and the plane is 1/ 2√3 . A constant horizontal force of 35 √ 3N is applied to the body causing it to 3 accelerate up the plane. Find the time taken for the body to reach the top and its speed on arrival
First, we need to find the force of friction acting on the body along the inclined plane. We can do this using the equation:
friction force = coefficient of friction * normal force
where the normal force is equal to the component of the weight of the body perpendicular to the plane, given by:
normal force = mg * cos(30)
where m is the mass of the body, g is the acceleration due to gravity, and cos(30) is the cosine of the angle between the weight vector and the perpendicular to the plane.
Therefore, the friction force is:
friction force = (1/2√3) * 5kg * 9.81m/s^2 * cos(30) = 5.10N
Next, we can resolve the constant horizontal force into its component forces parallel and perpendicular to the plane. The perpendicular force is equal to:
perpendicular force = 35√3N * sin(30) = 17.5N
This force is balanced by the normal force acting on the body. Therefore, we can focus only on the parallel force, which is equal to:
parallel force = 35√3N * cos(30) = 30.4N
This force is opposed by the force of friction. Therefore, the net force acting on the body is:
net force = 30.4N - 5.10N = 25.3N
Using Newton's second law of motion, we can find the acceleration of the body along the plane:
net force = ma
where a is the acceleration of the body.
Therefore, the acceleration of the body is:
a = net force / m = 25.3N / 5kg = 5.06m/s^2
We can now use the equation for uniformly accelerated motion to find the time taken for the body to reach the top of the plane:
s = ut + 1/2at^2
where s is the displacement of the body along the plane (equal to the length of the plane), u is the initial velocity (zero), a is the acceleration, and t is the time taken.
Rearranging, we get:
t = sqrt(2s / a) = sqrt(2 * 6.3m / 5.06m/s^2) = 1.78s
Finally, we can find the speed of the body on arrival at the top of the plane using the equation:
v = u + at
where v is the final velocity, u is the initial velocity (zero), a is the acceleration, and t is the time taken.
Therefore, the speed of the body on arrival at the top is:
v = 0 + 5.06m/s^2 * 1.78s = 9.01m/s (to 3 significant figures)
Well, this is quite a math-heavy question, but fear not, my friend, for Clown Bot is here to add a touch of humor to the situation!
First, let's tackle the time taken for the body to reach the top. You see, time is a funny thing. It tends to fly when you're having fun, but crawl when you're solving physics problems. So, let's see if we can speed things up a bit.
We can start by finding the net force acting on the body. The horizontal component of the applied force is 35 √ 3N, while the vertical component can be found by using some trigonometry and a dash of Pythagorean theorem. Once you have the net force, divide it by the mass of the body, and you'll have the acceleration.
Now, we can use the good old kinematic equation: final velocity equals initial velocity plus acceleration times time. Since the body is initially at rest, the initial velocity is zero. Rearrange the equation, plug in the numbers, and you'll find the time it takes for the body to reach the top.
As for the speed on arrival, well, it's always good to make a grand entrance, isn't it? So, why not calculate the final velocity when the body reaches the top? You can use the same kinematic equation, but this time the final velocity will be the one you're looking for. Calculate it, put on a top hat, and announce the speed with flair!
Remember, my friend, while physics problems can sometimes be a bit rough (just like that inclined plane), a bit of humor can go a long way in making them more bearable. Good luck with your calculations!
To find the time taken for the body to reach the top of the inclined plane and its speed on arrival, we need to follow the steps below:
1. Calculate the gravitational force acting on the body:
Gravitational force (Fg) = mass (m) * acceleration due to gravity (g)
= 5 kg * 9.8 m/s^2
= 49 N
2. Calculate the component of the gravitational force parallel to the incline:
F_parallel = Fg * sin(angle of incline)
= 49 N * sin(30°)
= 24.5 N
3. Determine the net force acting on the body:
Net force (F_net) = Applied force (F_applied) - Frictional force (F_friction)
= 35√3 N - F_parallel
4. Calculate the frictional force:
F_friction = coefficient of friction (μ) * normal force (Fn)
= (1/2√3) * Fg * cos(angle of incline)
= (1/2√3) * 49 N * cos(30°)
= 14.15 N
5. Calculate the net force:
F_net = 35√3 N - 24.5 N - 14.15 N
= 35√3 N - 38.65 N
= -3.65 N
Note: The negative sign indicates that the force is acting in the opposite direction.
6. Calculate the acceleration of the body using Newton's second law:
F_net = m * acceleration
-3.65 N = 5 kg * acceleration
acceleration = -0.73 m/s^2
7. Use the kinematic equation to find the time taken for the body to reach the top:
v = u + at
0 = 0 + (-0.73 m/s^2) * t
t = 0 s
Since the body starts at rest, the time taken to reach the top is 0 seconds.
8. Calculate the final velocity of the body at the top:
v = u + at
v = 0 + (-0.73 m/s^2) * 0 s
v = 0 m/s
The speed of the body on arrival at the top is 0 m/s.
To find the time taken for the body to reach the top of the inclined plane and its speed on arrival, we can break down the problem into several steps:
Step 1: Calculate the gravitational force acting on the body.
The gravitational force acting on the body is given by the formula Fg = m * g, where m is the mass of the body and g is the acceleration due to gravity. In this case, m = 5 kg and g = 9.8 m/s². Therefore, Fg = 5 kg * 9.8 m/s² = 49 N.
Step 2: Determine the component of the gravitational force parallel to the inclined plane.
Since the inclined plane is at an angle of 30° to the horizontal, the component of the gravitational force acting parallel to the inclined plane is Fg_parallel = Fg * sin(30°). Substituting the values, we get Fg_parallel = 49 N * sin(30°) = 24.5 N.
Step 3: Calculate the frictional force acting on the body.
The frictional force acting on the body is given by the formula F_friction = μ * F_normal, where μ is the coefficient of friction and F_normal is the normal force. The normal force is equal to the component of the gravitational force perpendicular to the inclined plane, which is given by F_normal = Fg * cos(30°). Substituting the values, we get F_normal = 49 N * cos(30°) = 42.43 N. Therefore, F_friction = (1 / (2 * √3)) * 42.43 N = 12.29 N.
Step 4: Calculate the net force acting on the body.
The net force acting on the body is the sum of the applied force, the component of the gravitational force parallel to the inclined plane, and the frictional force. Since the body is accelerating up the plane, the net force in this case is given by F_net = F_applied - Fg_parallel - F_friction. Substituting the values, we get F_net = 35√3 N - 24.5 N - 12.29 N = 35√3 N - 36.79 N.
Step 5: Use Newton's second law to find the acceleration.
According to Newton's second law, F_net = m * a, where F_net is the net force, m is the mass of the body, and a is the acceleration. Rearranging the formula, we get a = F_net / m. Substituting the values, we get a = (35√3 N - 36.79 N) / 5 kg = (35√3 N - 36.79 N) / 5 kg ≈ -0.616 m/s².
Step 6: Calculate the time taken for the body to reach the top.
To calculate the time taken, we can use the following kinematic equation: v = u + at, where v is the final velocity, u is the initial velocity (which is 0 m/s), a is the acceleration, and t is the time. Rearranging the equation, we get t = (v - u) / a. Since the body is initially at rest, the final velocity v is equal to the speed on arrival. Substituting the values, we get t = (v - 0 m/s) / -0.616 m/s².
Step 7: Calculate the speed on arrival.
We know that the final velocity v is equal to the speed on arrival. To find v, we can use the following kinematic equation: v² = u² + 2as, where v is the final velocity, u is the initial velocity (which is 0 m/s), a is the acceleration, and s is the distance traveled. Rearranging the equation, we get v = √(u² + 2as). Substituting the values, we get v = √(0 + 2 * (-0.616 m/s²) * 6.3 m).
So, to summarize:
- The time taken for the body to reach the top of the inclined plane can be calculated using t = (v - 0 m/s) / -0.616 m/s².
- The speed on arrival can be calculated using v = √(0 + 2 * (-0.616 m/s²) * 6.3 m).
Solving these equations will give you the exact values for the time and speed on arrival.