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√ . 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
To solve this problem, we need to break it down into two parts: finding the acceleration of the body up the inclined plane and then using that acceleration to find the time taken to reach the top and the final speed.
1. Finding acceleration up the inclined plane:
We can use the forces acting on the body to find the acceleration up the plane. The forces acting on the body are:
- Weight (mg) acting vertically downwards
- Normal force (N) acting perpendicular to the plane
- Friction force (f) acting parallel to the plane in the opposite direction to the motion
We need to resolve the weight and applied force into components parallel and perpendicular to the plane.
- Component of weight (mg sin 30) parallel to the plane = 5 x 9.8 x sin 30 = 24.5 N
- Component of applied force (35√3N) parallel to the plane = 35√3 x cos 30 = 30N
- Component of weight (mg cos 30) perpendicular to the plane = 5 x 9.8 x cos 30 = 42.4 N
- Normal force (N) = 42.4 N
- Friction force (f) = coefficient of friction (1/2√) x Normal force = (1/2√) x 42.4 = 7.4 N
The net force parallel to the plane = Applied force - Friction force - Component of weight
= 30 - 7.4 - 24.5 = -1.9 N (negative because it is acting in the opposite direction)
Using F=ma, we can find the acceleration up the plane:
- Net force parallel to the plane = mass x acceleration
- Acceleration = Net force parallel to the plane / mass = -1.9 / 5 = -0.38 m/s^2
2. Finding time taken to reach the top and final speed:
We can use the SUVAT equations (equations of motion with constant acceleration) to find the time taken to reach the top and final speed. We assume that the body starts from rest.
Using the SUVAT equation v^2 = u^2 + 2as, we can find the final speed (v):
- u (initial velocity) = 0
- a (acceleration) = -0.38 m/s^2 (negative because it is slowing down)
- s (displacement) = 6.3m (length of the inclined plane)
- v^2 = 0^2 + 2 x (-0.38) x 6.3
- v^2 = -4.788
- v = sqrt(-4.788) (taking the square root of a negative number gives an imaginary number, which means the body does not reach the top)
Since the body does not reach the top, we cannot use the SUVAT equations to find the time taken to reach the top. Instead, we can use the equation v = u + at to find the time taken for the body to come to a stop (v=0):
- u = 0
- a = -0.38 m/s^2
- t = (v-u)/a = (0-0)/-0.38 = 0 seconds
Therefore, the time taken for the body to come to a stop is 0 seconds.
Final answer: The body does not reach the top of the inclined plane and comes to a stop before reaching it. The time taken for the body to come to a stop is 0 seconds and its final velocity is 0 m/s.
To solve this problem, we can break it down into several steps:
Step 1: Find the gravitational force acting on the body.
Step 2: Determine the frictional force acting on the body.
Step 3: Calculate the net force acting on the body.
Step 4: Use Newton's second law to find the acceleration of the body.
Step 5: Find the time taken for the body to reach the top of the inclined plane.
Step 6: Calculate the final velocity of the body at the top of the inclined plane.
Let's follow these steps one by one:
Step 1: Find the gravitational force acting on the body.
The gravitational force acting on the body is given by the formula:
Gravitational force = mass * gravitational acceleration
Gravitational acceleration is approximately 9.8 m/s^2.
Gravitational force = 5 kg * 9.8 m/s^2 = 49 N
Step 2: Determine the frictional force acting on the body.
The frictional force is given by the formula:
Frictional force = coefficient of friction * normal force
The normal force is equal to the gravitational force acting on the body.
Normal force = gravitational force = 49 N
Frictional force = (1/2)√ * 49 N
Step 3: Calculate the net force acting on the body.
The net force is the difference between the applied horizontal force and the frictional force.
Net force = Applied force - Frictional force
Applied force = 35√3 N
Net force = 35√3 N - (1/2)√ * 49 N
Step 4: Use Newton's second law to find the acceleration of the body.
Newton's second law states that the net force is equal to the mass of an object multiplied by its acceleration.
Net force = mass * acceleration
Acceleration = Net force / mass
Acceleration = (35√3 N - (1/2)√ * 49 N) / 5 kg
Step 5: Find the time taken for the body to reach the top of the inclined plane.
We can use the kinematic equation to find the time taken for the body to reach the top.
vf = vi + at
Since the body starts with an initial velocity of 0 m/s, the equation simplifies to:
vf = at
The final velocity at the top of the inclined plane is 0 m/s, so we can rewrite the equation as:
0 = a * t
t = 0 / a
The time taken for the body to reach the top is 0 seconds.
Step 6: Calculate the final velocity of the body at the top of the inclined plane.
The final velocity at the top of the inclined plane can be found using the following equation:
vf = vi + at
vf = 0 + (35√3 N - (1/2)√ * 49 N) / 5 kg
This is the final velocity of the body at the top of the inclined plane.
To find the time taken for the body to reach the top and its speed at the top of the inclined plane, we'll need to break down the problem into multiple steps. Here's an outline of the steps:
1. Calculate the gravitational force acting on the body.
2. Calculate the frictional force acting on the body.
3. Calculate the net force acting on the body.
4. Calculate the acceleration of the body.
5. Use kinematic equations to find the time taken for the body to reach the top and its speed at the top.
Let's go through each step in detail:
Step 1: Calculate the gravitational force:
The weight of an object can be calculated using the formula:
Weight = mass × acceleration due to gravity
Given that the mass of the body is 5 kg, and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the weight of the body:
Weight = 5 kg × 9.8 m/s²
Step 2: Calculate the frictional force:
The frictional force can be found by multiplying the coefficient of friction (μ) by the normal force. The normal force is the component of the weight perpendicular to the inclined plane. In this case, it would be equal to:
Normal force = Weight × cos(theta)
where theta is the angle of inclination (30°).
Frictional force = coefficient of friction × Normal force
Given that the coefficient of friction (μ) is 1/2√2, we can now calculate the frictional force.
Step 3: Calculate the net force:
The net force acting on the body is the difference between the applied force and the frictional force. The applied force is given as 35√3 N.
Net force = Applied force - Frictional force
Step 4: Calculate the acceleration:
Using Newton's second law of motion, we can relate the net force and acceleration using the formula:
Net force = mass × acceleration
Rearranging the formula, we get:
Acceleration = Net force / mass
Step 5: Calculate the time taken and speed at the top:
We need to use kinematic equations to find the time taken for the body to reach the top and its speed at the top. The relevant equation is:
vf^2 = vi^2 + 2as
where
vf = final velocity (speed at the top)
vi = initial velocity (0 m/s as the body starts from rest)
a = acceleration (calculated in step 4)
s = displacement (distance travelled up the inclined plane)
Since the body starts from rest, the initial velocity (vi) is 0. We know the acceleration (a) from step 4. The displacement (s) is given as the length of the inclined plane, which is 6.3 m.
Now we can plug these values into the equation and solve for vf. Once we have vf, we can use the equation:
vf = vi + at
to find the time taken (t) for the body to reach the top.
I hope this breakdown helps you in understanding how to solve the problem. Let me know if you have any further questions!