A force F = (4.0 N)i + cj acts on a particle as the particle goes through displacement d = (3.4 m)i - (2.0 m)j. (Other forces also act on the particle.) What is the value of c if the work done on the particle by force F is each of the following?
(a) zero
(b) 19 J
(c) -17 J
Work is the dot product of the Force and Dispacement vectors.
Set W = 13.6 - 2c and solve for the c that goes with each W.
(13.6 -2c is the dot product.)
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(a) c = 0
(b) c = 8.5
(c) c = -9.5
To find the value of c for each case, we need to calculate the dot product between the force and displacement vectors and set it equal to the given work values.
(a) When the work done on the particle is zero:
The dot product of two vectors is zero when they are perpendicular to each other. Therefore, we can set up the equation:
(4.0 N)(3.4 m) + c(-2.0 m) = 0
Simplifying this equation gives:
13.6 - 2c = 0
Solving for c, we get:
-2c = -13.6
c = 6.8
(b) When the work done on the particle is 19 J:
We can set up the equation:
(4.0 N)(3.4 m) + c(-2.0 m) = 19 J
Simplifying this equation gives:
13.6 - 2c = 19
Solving for c, we get:
-2c = 19 - 13.6
-2c = 5.4
c = -2.7
(c) When the work done on the particle is -17 J:
We can set up the equation:
(4.0 N)(3.4 m) + c(-2.0 m) = -17 J
Simplifying this equation gives:
13.6 - 2c = -17
Solving for c, we get:
-2c = -17 - 13.6
-2c = -30.6
c = 15.3
Therefore, the value of c for each case is:
(a) c = 6.8
(b) c = -2.7
(c) c = 15.3
To find the value of c in each case, we need to use the formula for work done by a force:
W = F ⋅ d
To calculate the dot product, we multiply the x-components of F and d, and then add it to the product of the y-components of F and d.
Let's calculate the dot product for each case:
(a) When the work done is zero:
W = 0
Using the formula, we have:
0 = (4.0 N)(3.4 m) + c(-2.0 m)
Simplifying, we get:
0 = 13.6 N⋅m - 2c m⋅N
Since work is zero, the dot product must be zero. Therefore, we can set 13.6 - 2c = 0:
13.6 - 2c = 0
2c = 13.6
c = 6.8
So, when the work done is zero, c = 6.8.
(b) When the work done is 19 J:
W = 19 J
Using the formula, we have:
19 J = (4.0 N)(3.4 m) + c(-2.0 m)
Simplifying, we get:
19 = 13.6 - 2c
Rearranging the equation:
-2c = 19 - 13.6
-2c = 5.4
c = -2.7
So, when the work done is 19 J, c = -2.7.
(c) When the work done is -17 J:
W = -17 J
Using the formula, we have:
-17 J = (4.0 N)(3.4 m) + c(-2.0 m)
Simplifying, we get:
-17 = 13.6 - 2c
Rearranging the equation:
-2c = -17 - 13.6
-2c = -30.6
c = 15.3
So, when the work done is -17 J, c = 15.3.
Therefore, the values of c for each case are:
(a) c = 6.8
(b) c = -2.7
(c) c = 15.3